Talk:Euclidean vector/Archive 5

Is Vector a kind of Tuple?
Well... is it?

I think it is, but I have no formal background in mathematics, so I will not put it in.

If it is, I wish someone would add this to the description, because it eases generalization of mathematical concepts, which is a pretty neat thing. —The preceding unsigned comment was added by 200.164.220.194 (talk) 02:53, 23 December 2006 (UTC).


 * Mathematically speaking, a vector can be represented by a tuple. See Coordinate_vector for more information.192.55.12.36 (talk) 21:33, 22 April 2008 (UTC)


 * Thanks for drawing my attention to this thread. See tuple.  I am adding it to the disambiguation page vector.  silly rabbit  ( talk ) 21:23, 22 April 2008 (UTC)

Vector Symbols
Can anyone tell me what the difference between having the arrows above and below the vector means? i.e. $$\overrightarrow{AB}$$ and $${CD}_{\rightarrow}$$ -- pizza1512 Talk Autograph 18:10, 30 April 2007 (UTC)


 * The overarrow is the "American" style, the underarrow the "European" style. I use both, so I don't know what that makes me. Silly rabbit 06:27, 25 May 2007 (UTC)

Isn't the symbol $$\overrightarrow{AB}$$ a symbol for a ray? Professor Calculus (talk) 15:57, 6 March 2008 (UTC)


 * The ray from point A to point B is equivalent to a vector with the same slope and magnitude as the ray. In this context, we usually think of the vector as the parallel transposition of the ray, with A mapping to the origin of the coordinate system. I use $$\overrightarrow{X}$$ to distinguish the vector X from the variable X, and use $$\overrightarrow{XY}$$ for the ray from point X to point Y, myself. Pete St.John (talk) 18:07, 6 March 2008 (UTC)


 * Oh, thanks. I didn't realize that.Professor Calculus (talk) 00:51, 7 March 2008 (UTC)

For the representation of vectors perpendicular to a page, it is pretty clear that circle-dot (&#x2299;) is coming up from the page, however for going down into the page, is it circle-x (&otimes;) or circle-cross (&oplus;)? The text says the latter but the big image shows the former. I think the text is right because the circle-dot is also a symbol for the Sun (hence up), while the circle-cross is also a symbol for the Earth (hence down). --George Hernandez (talk) 20:02, 17 April 2008 (UTC)


 * It's always a circle with an x in it, not a + or cross. The text should be corrected. --Steve (talk) 17:02, 18 April 2008 (UTC)

Cleanup - What is this article about?
Silly rabbit asks that I list criticisms of this article to assist with a rewrite that he is planning. I had been thinking of waiting until the AFD of Vector (physical) was over but it's good to jot these points down while they are fresh in my mind.


 * Starting at the top, we have the title, for which there is some previous discussion. The term spatial vector does not seem right.  Its usage seems to occur in the specialist fields of ECG analysis and  soliton waves.  Feynman talks of a space-vector which is plainer English but begs the question of what sort of space we are talking about - the general reader might suppose this is Outer space.  In a case like this, where the qualifying word in parentheses indicates a domain of knowledge or context, the title Vector (mathematics) might be better.


 * If we look at the content, we see that the article does not just describe vectors and their representation, but also includes simple operations like scalar multiplication and vector addition. In some cases, these topics are redirected to this article, e.g. Vector sum.  My preference would be to break these operations out into separate articles as they seem to overload this one.  But if they are retained, they indicate that the title perhaps ought to be broader - what Feynman calls vector algebra.


 * The opening sentence uses the word object which is a poor choice when discussing an abstraction. A fundamental concept of this sort should be explained with more care and the lede should be polished word-by-word.


 * The article lacks good examples and there is some previous discussion of this too. There is one example of a velocity of 5 metres per second upwards and the notation (0,5) is used for this.   This seems poor in that it does not explain the notation, the origin and the use of just two dimensions.  And there is no discussion of the fact that your upwards is not the same as mine, because we are standing on a sphere in a gravitional field.  Oversimplification of this sort, which is not well-linked to the physical world, will tend to confuse the general reader.  I would like to see at least one genuine well-developed example - perhaps something from aeronautics where they talk of thrust vectors and intercept vectors.


 * The article is poorly sourced. A good general guideline is that there should be a citation to a reliable source per paragraph.


 * Wikipedia is not a textbook and so we should be careful to avoid the exact style of works which are, such as Feynman's lectures which I allude to above. Also, we are not trying to impress but inform and should consider that our readership is the world.  When I was young, I used to enjoy reading works such as the Children's Encyclopedia and so I suppose that our readership includes bright children of age 10 or younger.  The article currently seems too dry for this readership.  For example, the original text used the  word arrow which seems natural.  The article now uses the term line segment.  This perhaps improves the mathematical rigour but that's not good if causes people to stop reading.

Note that I criticise no particular editor. The previous editors are to be congratulated upon having got the article to this point.

Colonel Warden (talk) 08:22, 1 March 2008 (UTC)


 * I agree that the title is suboptimal. However, before anything else we first need to agree on what the article is about.  The original point of this article, as I understand it, was to be about the concept that is informally introduced (at a freshman level) as something with "magnitude and direction", and is formalized at a more advanced level (defining what it means to "have a direction") as being a vector space contravariant with spatial coordinates under rotations.


 * (Such spaces are always finite-dimensional in practice, and always have a dot product...they need one in order to define the rotation group, and also need one to define the "magnitude" of the vector...and being concepts in differential geometry they also have two-forms, or cross products in 3d. And all vector spaces have addition and multiplication by scalars&mdash;if you don't have these, it's not a "vector" by any definition in mathematics or physics, so your suggestion that these don't belong in the article does not seem right to me.)


 * Vector (mathematics) is not right unless we are talking about general vector spaces. There is no need for such an article, since we already have vector space; also, this concept is considerably more abstract than the "magnitude and direction" notion of vectors (e.g. it includes infinite-dimensional vector spaces and vector spaces not over the real or complex numbers), and need not include a dot product (a dot product, or at least a norm, is necessary to have a "magnitude").


 * Vector (physics) is not right either, as there are many types of vector spaces used in the physical sciences, both finite and infinite-dimensional and both contravariant and more abstract.


 * Vector (contravariant) would be correct, but would be off-putting and unfamiliar to undergraduate students who should otherwise be the target readers for an article on this subject; it also might be a bit too general&mdash;we should probably stick to contravariance in Euclidean/Cartesian space under rotations, and leave curvilinear coordinates, curved manifolds, and different groups (e.g. the Lorentz group in relativity) to other articles. Vector (polar) is another term for this concept, but it is even more obscure and will easily be confused with polar coordinates. Since we are primarily talking about contravariance with respect to the rotation group in three dimensions, in relativity they would occasionally be called "three vectors" in contrast to "four vectors" which are contravariant with respect to the Lorentz group in spacetime, but this terminology is also unfamiliar to most users of such vectors.


 * Firefly suggested Vector (Gibbs-Heaviside) in reference to the historical originators of this concept of vectors (although the precise notion of contravariance didn't come later, as I said this is just a formalization of the earlier notion of "having a direction"). However, this terminology does not seem widespread.


 * Vector (magnitude and direction) is a bit wordy, but would be reasonably clear and uses the most common (albeit informal) terminology used to describe the concept. Vector (spatial) was an attempt at an abbreviated reference to the fact that contravariant vectors have a specific relationship to spatial positions (hence "direction"), but is not a widespread terminology.  My vote, as the best compromise title I can think of, would be for Vector (magnitude and direction), unless someone can come up with a better suggestion (or unless we are going to change the subject of this article entirely).


 * (References are not a problem to add. There are dozens (at least!) of undergraduate calculus and physics textbooks covering this stuff at a basic level.  And there are plenty of books covering the definitions at a more sophisticated level; e.g. Arfken &amp; Weber, Mathematical Methods in Physics has a particularly clear discussion without going too far into differential geometry.)


 * —Steven G. Johnson (talk) 17:14, 1 March 2008 (UTC)


 * I'm surprised that you see this as an undergraduate/freshman topic. The basics of vectors are introductory material for science/maths at a secondary level in the UK and so pupils will meet this at age 13 or so.  I suppose that it is much the same in the USA, i.e. high school.  No?  Colonel Warden (talk) 19:59, 1 March 2008 (UTC)


 * It's a while since I did my O-levels, but I'm pretty sure we didn't cover vectors as directed lines segments (A-level), vector addition (also A-level) and scalar products (that would be A-level again). What we did cover was that vectors have direction and magnitude, and that velocity, acceleration etc were vectors.  I find it difficult to believe that today's 13-year-olds are doing stuff I did at A-level.  However, I do believe it is the case that some aspects of mathematics (calculus, for example) which US students only meet at the age of 18 are on current A-level syllabuses; it's possible these include the basics of vectors. --Sturm 20:46, 1 March 2008 (UTC)


 * In the UK, the current National Curriculum specifies the following at Key Stage 4 for Mathematics which is ages 14-16:


 * ''Vectors


 * ''understand and use vector notation; calculate, and represent graphically the sum of two vectors, the difference of two vectors and a scalar multiple of a vector; calculate the resultant of two vectors; understand and use the commutative and associative properties of vector addition; solve simple geometrical problems in 2-D using vector methods.


 * The equivalent stage for Science is less clear on this subject but pupils are expected to understand "the difference between speed and velocity".


 * Now these are core subjects and so this is universal education that everyone is supposed to get (though many fall by the wayside, of course). Wikipedia should be operating primarily at this level since our readership is universal.  When Steven says that this is a college-level topic, we have a significant difference of expectation.  The UK has developed a National Curriculum to make such expectations clearer but most academic institutions are familiar with the concept of a syllabus.  That's what Wikipedia needs perhaps - some method of breaking topics into introductory, intermediate and advanced levels.  Colonel Warden (talk) 22:05, 1 March 2008 (UTC)


 * What you've just outlined appears to be from the key stage 4 higher programme of study "Ma3 shape, space and measures". Higher programmes are recommended to be taught to students who've achieved "a secure level 5" or level 6 at key stage 3.  They aren't recommended to be taught to those achieving lower than this, so I'm not seeing this as a "core subject". (And that's without going to the issue of those who are "taught" but still arrive at university without understanding.)   --Sturm 23:00, 1 March 2008 (UTC)


 * You're right, this is covered in high school too in the US, to a limited extent. Certainly, the article should begin by being as accessible as possible, although farther down in the article it can have subsections on more advanced concepts.  In any case, both the high school and the freshman level informal definitions are the same (things with "magnitude and direction"), and the question becomes where to go from there.  You have to talk about addition of vectors and multiplication by scalars, both of which are fundamental to the concept of "vector", and you have to talk about dot products (to get a magnitude).  And you have to give at least some indication of how these informal concepts connect to more precise and/or general notions of vectors.  In particular, at some point the article needs to say precisely what it is about, what "magnitude and direction" really means, and what types of  vector concepts are included (so that the article isn't constantly being tugged into different directions from different generalizations of the vector concept).   —Steven G. Johnson (talk) 22:26, 1 March 2008 (UTC)

What is this article about?
I think the question which Steven Johnson raises above is a good one, and probably deserves to be addressed in its own section. As a model for the original article, I used the introduction and first chapter of Pedoe's Geometry, in which the "magnitude and direction" aspect of spatial vectors is mentioned, although not necessarily emphasized to the exclusion of all other points of view. Pedoe treats vectors as synthetic geometric objects on their own: directed line segments (or equivalence classes thereof). The Encyclopedic Dictionary of Mathematics and the Springer Encyclopedia of Mathematics also take this approach. (Curiously, the Springer article is called vector (geometric), which may be another possible title.) I found the approach quite appealing because it readily permits the idea of spatial vector to be discussed outside of a purely Euclidean context. Thus a vector in special relativity, as a directed line segment, is on equal footing with a vector in classical mechanics.

The idea of "magnitude and direction" is a good one to guide intuition, and for the purposes of explanation, but as Steven points out, it is rather poor as a definition. Looking again at Tom Apostol's Calculus, he seems to be aware of issues such as these, and identifies three distinct approaches:


 * The analytic approach: Essentially do everything in coordinates.
 * The geometric approach: Directed line segments
 * The axiomatic approach: Vector spaces.

I believe that the consensus is that other articles satisfactorily address the third of these possibilities, and that this article should do no more than perhaps point the way to them.

I think that this article should handle both the geometric and the analytic approaches to spatial vectors. There are some arguments against this, I realize. The chief among them is perhaps that there already are articles on the analytic concept of vectors (e.g., coordinate vector), and that, furthermore, this article will never be able to break its current duality unless it selects one definite approach to take. My replies to these objections are as follows. Firstly, none of the existing articles (coordinate vector, vector calculus, etc.) appears quite ready to take up the torch of presenting analytic vectors in a coherent and well-motivated way. Furthermore, modern readers are likely to see both concepts developed in tandem rather than separately, and I think the vector (spatial) article should keep with this tradition. Secondly, when one gets into the physics (or deeper into the geometry), the coordinate approach and the geometric approach become inextricably linked. In physics, one deals with the ideas of covariance under changes in the coordinates, and in geometry with similar notions via the Erlanger program. In each case, the coordinates are not what one uses to model the vector per se (they are not an "intrinsic feature" of the vector's description), but rather are a tool for modeling the space around the vector. Thus, in physics and geometry, one thinks of the vector as existing independently of the coordinate system. Hence I believe that the geometric approach is the one which must be given precedence in the article. Silly rabbit (talk) 18:37, 1 March 2008 (UTC)


 * Even restricting considerations to the "analytic" approach, there is a tension between the definition of a "vector" as essentially anything in $$\mathbb{R}^n$$, and the physics/differential-geometry notion that "having a direction" implies a specific relationship to some notion of position (i.e. contravariance). The latter approach is taken e.g. in Arken & Weber, Mathematical Methods in Physics, although they also describe abstract vector spaces as the alternative.


 * Put another way, starting from the informal concept of "magnitude and direction", there are essentially two routes one can follow towards more advanced and precise concepts. One is the concept of a Banach space (not just a vector space if you want to have a "magnitude"), and the other is the concept of contravariant vector spaces, tensors, and differential geometry (which formalize the notion of "direction").  The article, at least, needs to make it clear that both of these routes are possible, and distinct choices, and needs to make it clear how far along either route we intend to go in this article.


 * The typical physics point of view (as e.g. in Arken & Weber) is that contravariance is the natural formalization of the specific concept of "magnitude and direction", whereas abstract vector/Banach spaces are a more general concept (that includes contravariant vectors as a special case).


 * —Steven G. Johnson (talk) 19:16, 1 March 2008 (UTC)


 * Yes, that is another issue that I had attempted to address during an earlier incarnation of this article: how and when to introduce "magnitude" and "direction". I didn't get very far beyond consolidating all discussion of the dot product and norm into a section of its own, and miscellaneous other movement of text around.  I pretty much gave up trying to make a decent "overview," but perhaps more editors can now come up with something better.  (It's sort of a shambles now, I know, but it is significantly better than how it started out about a year and a half ago.)


 * Magnitude and direction can both be built into a transformation-group (Erlangen or covariance-based) approach to vectors, and so have appropriate generalizations to basically any geometry. Two vectors have the same "magnitude" if they are congruent under the action of the Euclidean group;  they have the same direction if they are congruent by a scaling-translation.


 * I was reluctant to ascribe "magnitude" a particular mathematical meaning in the beginning of the article, but it seemed impossible to abandon the term completely. The Encyclopedic Dictionary cleverly dodges the problem by saying something like: "The concept of vector originated with the physical notions of velocity and force, which are quantities having both a magnitude and a direction."  The article then goes on to treat vectors geometrically until enough background has been developed to introduce magnitude safely.


 * However, you now raise yet another interesting point. As long as one is willing to grant that magnitude means length, then various sorts of normed spaces and inner product spaces are natural generalizations.  Even in finite dimensions, it is trivial to write down norms which are not the usual Euclidean length.  However, these more general spaces are probably never associated with the idea of a spatial vector, and this article should do little more than indicate the existence of such generalizations, and provide appropriate links.  Silly rabbit (talk) 19:49, 1 March 2008 (UTC)


 * The persistent problem with this article is that it gets tugged in too many directions, with too many possible generalizations and notions of "vector".


 * I really think that this article should as much as possible stick to the elementary concept of a vector as things with magnitude and direction, with the specific meaning of "magnitude" in terms of the ordinary norm, with the usual operations on such vectors, and some typical applications, with a short section saying that "direction" can be formalized in terms of contravariance, and with another short section giving brief pointers to other articles on different generalizations.


 * The audience for this article should be people who are trying to understand "vectors" as they are used in elementary calculus and physics courses, and it should therefore only give brief pointers to other articles for generalizations that are not used in such contexts. (Contravariance/covariance, on the other hand, is not a generalization so much as a precise definition of "having a direction".)


 * —Steven G. Johnson (talk) 20:07, 1 March 2008 (UTC)


 * Yes, I basically agree with you. I wasn't advocating including a bunch of generalizations in the early parts of the article, but was rather trying to rationalize the definition of a vector as a "directed line segment" as being the geometrically correct one.  We can still have magnitude and direction, of course, but I don't think that should be advanced as the definition since it is inadequate for a variety of different reasons. Silly rabbit (talk) 20:31, 1 March 2008 (UTC)
 * This seems unnatural to me. How is a velocity, or an electric field, or a gradient, a "directed line segment"?  (Who defines it this way, except for displacement vectors?)  My suggestion would be to give the "magnitude and direction" as the informal definition (this is totally standard, almost universal in introductory courses), and say that this can be formalized and made precise in a variety of ways&mdash;the standard way (in physics and differential geometry, when talking about things with "magnitude and direction" as opposed to other abstract types of vector spaces) being as contravariant vectors&mdash;whereas there are also abstract generalizations of "vector" that aren't naturally described as having either a "direction" or a "magnitude" (i.e. a norm) and which are only mentioned in the article.  —Steven G. Johnson (talk) 21:57, 1 March 2008 (UTC)
 * Several sources define it this way. The sources I have been looking at include the Springer Encyclopedia of mathematics, the Encyclopedic Dictionary of Mathematics, Dan Pedoe's Geometry, and Tom Apostol's Calculus.  These are all fairly standard mainstream sources.  I think the first two are especially pertinent because they are from encyclopedias, which must strive to give an encyclopedic definition.  However, I also understand your objection to the arrow definition, that it perhaps is not encompassing enough to all quantities which would be relevant to physicists (which "have a magnitude and direction").  If one considers space as the physical position space, then indeed velocity, force, etc., are not arrows in that space but instead in some other auxiliary Euclidean space (a tangent space or other such).  That's obviously treading into untenable territory, and should be avoided as well.  This seems to be the source of the duality in the article, then: neither approach is correct.  One can either give a suitable (and reasonably rigorous) definition of a geometrical vector, or a hand-waving "magnitude and direction" characterization which covers the uses in physics but is wholly inadequate as a definition.  Silly rabbit (talk) 22:16, 1 March 2008 (UTC)

(restarting indenting)

The problem is, any definition which appears to exclude velocity as a "vector" is going to alienate a large fraction of the audience for this article (people who learned, or are learning, these concepts in the context of high-school/undergraduate math and science courses). What's wrong with handwaving about "magnitude and direction" (calling it an "informal" definition) while stating that there is a more precise definition as well as other generalizations, and then giving the more precise definition of "direction" via contravariance (in the simple case of Cartesian coordinates under rotations) for advanced readers in a subsection? (You don't need to go into tangent spaces as long as you're willing to represent vectors by components.)

This is not really any different from, say, real number. You neither want nor need to start with any rigorous definition in terms of Dedekind cuts etc.; you start with an informal notion, and then eventually give a precise meaning for those (relatively few) readers who will be sophisticated enough to appreciate it.

—Steven G. Johnson (talk) 22:34, 1 March 2008 (UTC)


 * I can think of a few problems with putting "magnitude and direction" ahead of other definitions. The first of which is that the article really should have a definition, if possible, as early on as possible.  Readers shouldn't have to wait until the last few sections.  Secondly, the magnitude and direction characterization, while immensely useful for saying what things in physics can be described by vectors, does not say what can be done with vectors.  For instance, to add two vectors, you either need to commit to a coordinate system (which I think should be avoided if possible), or you need to represent them as arrows in a Euclidean space so that you can apply the parallelogram rule.  Perhaps we can come to some suitable language which will encompass both points of view. Silly rabbit (talk) 22:44, 1 March 2008 (UTC)

A possible new intro section
Just to be more concrete, let me propose a possible new introduction:


 * In elementary mathematics and physics, a vector is informally defined as something described by a magnitude (a non-negative number) and a direction. Geometrically, vectors  are often represented by directed line segments ("arrows") with length proportional to the vector magnitude and pointing in the direction of the vector.  A typical example is velocity, which has both a magnitude (the speed, e.g. 100 km/h) and a direction (e.g. "north"), and might be represented geometrically by an arrow pointing in the direction of motion with a length proportional to the speed.  Since its earliest mathematical formulation by Gibbs and Heaviside in the 1880s, this vector concept has become widely used in science and engineering to describe numerous physical variables (such as velocity, force, and electric fields) and also forms the foundation of vector calculus.


 * This article is about this basic notion of vectors with magnitudes and directions: their applications, the standard operations on such vectors, and their typical representations either geometrically or in terms of vector components. However, the subject of vectors leads towards many more advanced and abstract concepts, which are described in more detail by other articles.


 * First, the informal concept of "having a direction" is made precise, in physics and differential geometry, by defining a specific relationship, called contravariance, between the vectors (more precisely called contravariant vectors) and a separate notion of spatial "positions." The simplest definition of contravariance, given below, is for Cartesian coordinates in ordinary Euclidean space under rotations, but in differential geometry this is extended to curved manifolds via tangent spaces.   The concept of contravariant vectors can be further generalized to tensors, as well as to vectors contravariant with more than spatial positions (e.g. four vectors in special relativity, which are contravariant with both space and time under Lorentz transformations).


 * Second, there are also much more abstract generalizations of the "vector" concept that are not directly associated with "directions" or "magnitudes," but which share the operations of vector addition and rescaling: vector spaces. In an abstract vector space, the vectors may not have a magnitude (length); the most common abstract definition of vector spaces that do include vector "magnitudes" is called a Banach space.  In this viewpoint, contravariant vectors form only one specific class of Banach spaces, in which the vectors have a precise relationship to spatial "directions."

Comments? Besides nitpicks over individual phrasings, is the overall structure of the intro and the definition of the article topic something we can mostly agree on? (Note the important distinction between saying that a vector is a directed line segment versus saying that a vector can be represented by a directed line segment.)

(The brief discussion here of more advanced topics is necessary, both to give readers interested in such topics pointers of where to go, and to clearly demarcate the subject of this article from other articles on Wikipedia...the history of this article shows that, without such a clear up-front demarcation and an explanation that there is a formal meaning of "having a direction," this article will descend into confusion every time an editor who is a math major comes across it. At the same time, by clearly stating that the last two paragraphs are about advanced topics, I think we can avoid intimidating elementary readers.)

Once we agree on the subject of the article, then it will be easier to discuss the title.

—Steven G. Johnson (talk) 05:56, 2 March 2008 (UTC)

I do not like the proposed version, and prefer the current one, despite its perceived lack of applicability to physics. Here are a few comments: Just my two cents. Silly rabbit (talk) 15:04, 2 March 2008 (UTC)
 * 1) This one may be a bit nitpicking, but why is it legitimate to say that velocity (etc.) are vectors, whereas directed line segments only represent vectors.  If anything, this conflates the noumenon (a vector) and phenomenon (the object of human measurement: velocity).  I think the line segments are vectors, and velocity (etc.) are represented as vectors.  For an object to have a magnitude and direction (in the sense agreed to above) means that the object lives in some Euclidean space, and is a directed line segment there.  This is also the meaning of the complicated covariant description of vectors, whether you believe it or not (see Klein geometry).  I think, contrary to your suggestion, we can be upfront with a reasonably rigorous definition without descending into confusion as long as it is done properly.  At the moment, however, it seems that we are unable to break the stalemate on the issue of the definition of a vector.  Some outside input might be helpful.
 * 2) I think the history is wrong.  Gibbs and Heaviside did not originate the idea of a spatial vector.  That honor should probably be given to Hamilton who certainly thought of his quaternions in much the same way as our spatial vectors.  Use of the word "vector" in its modern form can certainly be found in the writings of Clifford in the early 1870s, though I am not certain if he was the first to do so.  Vectors as lists of numbers are conventionally credited to JJ Sylvester, but these are not "spatial" vectors.  Certainly Hermann Grassmann's 1844 work bears mentioning (although no-one seems to have read it until vectors were already commonplace).  (Note: Someone really should write a history section.)
 * 3) It is too long, and spends too much time on things which are barely mentioned in the article.  Three paragraphs (!) are spent detailing the generalizations of the subject of the article, rather than discussing the subject itself.  My original intent (about a year ago) was to move the information about generalizations, covariance, and so forth out of the lead and into the "Overview" section.  Certainly, a brief mention of covariance can be given, but it should be proportional to its relative treatment in the article.
 * 4) The remaining parts of the proposed lead do very little to summarize the content of the article.  The lead should at least mention coordinates (in a way that will be understandable to someone who doesn't know what a "rotation" is).  It should also mention the fundamental operations defined on vectors and, I feel, say what those operations mean.  The bulk of the article is dedicated to such things, and they need to be specifically enumerated in the lead, per WP:LEAD.
 * 5) One thing the new version does quite unambiguously is to specify the scope of the article.  It effectively says that the article is not going to do anything rigorously, and that other, more rigorous definitions can be found in other articles.  I suspect that this is overcompensation for my philosophical discussions above regarding the current structure of the article, and definition of a vector as a directed segment in Euclidean space.  However, I do not believe that rigor needs to be sacrificed for clarity in this case.


 * Regarding your points, in order.


 * First, the important thing is not your opinion, or my opinion, it is what is in standard references. And there are many, many texts and articles etc. that say that velocity is a vector.  This is standard usage of the term, philosophical quibbles aside.  (This reminds me of the endless discussion of whether a vector can be described as a "quantity"....the point is that people do use the word this way in published references, whether or not you agree with it.)


 * Again, with regard to history, most sources seem to say that the modern formulation of vectors is due to Gibbs and Heaviside. Certainly, the terminology and the enumeration of the operations is due to them; quaternions were expressed fairly differently (with a "vector part" and a "scalar part", both of them were considered parts of the same thing).  Certainly, we can have a history section that goes into the history in more detail, and discusses the antecedents of the modern formulation.


 * Actually, only two paragraphs are spend discussing generalizations. Do we count differently?  And it seems kind of important to me for the introduction to explain what the topic of the article is (rather than to fit in all of the content of the article), and how it relates to other concepts of "vector" in math and physics.


 * It does mention coordinates. Defining what they mean will take too long for the introduction and has to be left to the text within the article, as will defining all the standard operations (addition, scaling, subtraction, dot products, cross products).  (And please don't lecture me that cross-products are two-forms...at this level, vectors are at most three-dimensional and cross-products are treated as one of the standard vector operations.)


 * Actually, the introduction does say that a rigorous definition of contravariant vectors, at least for Euclidean space, will be given in the article. (This is fairly easy to do at an undergraduate level if you are willing to work in terms of components.)  Honestly, though, most readers will have no use for that level of rigor, which is why it should go in only one subsection.  Nor is that level of rigor typically found in elementary treatments of this subject.  Nor should this article be about abstract vector spaces.  Nor, honestly, is it all that "un-rigorous" to define the operations both geometrically and in terms of components...are you confusing "rigorous" with "abstract and general"?


 * —Steven G. Johnson (talk) 16:15, 2 March 2008 (UTC)


 * I think you've missed the point, at least the first one. There may be some vector we call "velocity" and it may represent some particular magnitude and direction of motion (in some inertial frame of reference), but it is called velocity because it represents what is actually velocity. If I had a function P(X) which represents my profit, what is my profit? I might say "P(X)" but it isn't really profit - it's a function. Profit is money in the bank. P(X) just happens to share the name "profit" because it represents profit - by the same token, the money in my bank is certainly not a function - P(X) is the function. Likewise, "velocity" may be represented by a vector, but it is not a vector. A vector is an abstraction. Vectors do not exist in real life. --Cheeser1 (talk) 16:24, 2 March 2008 (UTC)

Regarding the history, is it really true that most sources agree that the modern formulation of vectors is due to Gibbs and Heaviside? Granted, they are the ones traditionally credited with the assembly of the subject currently known as vector analysis. But the conception of vectors as objects with a magnitude and direction clearly predates their work. See for instance Clifford, Preliminary sketch of biquaternions, Proceedings of the London Mathematical Society 1871 s1-4(1):381-395. A direct quote:
 * The vectors of Hamilton are quantities having magnitude and direction, but no particular position; the vector AB being regarded as identical with the vector CD when AB is equal and parallel to CD and in the same sense.

This is precisely the modern conception of a spatial vector. Silly rabbit (talk) 16:32, 2 March 2008 (UTC)
 * For anyone whose English, particularly historical-mathematical-English, might be lacking, "equal" means equal magnitude, "parallel" means what it normally means, and "same sense" means pointing the same direction (since "parallel" could mean same direction or directly opposite). --Cheeser1 (talk) 16:39, 2 March 2008 (UTC)
 * And for those who like to use more concise terminology, "codirectional" means "having the same direction", that is "parallel and in the same sense". Paolo.dL (talk) 15:48, 19 August 2008 (UTC)

Another point, which I may not have explained clearly enough, is that I regard it as perfectly rigorous to define a vector as a directed line segment in Euclidean space. One can then proceed to also define the addition, subtraction, and scalar multiplication in this context. That is what I mean when I say "rigorous": Rather than using a non-definition ("magnitude and direction") let's at least advance a definition ("arrow in Euclidean space"). In fact, although Steven may not feel it is suitably rigorous or encompassing of everything he would like to call a vector, it turns out to mean precisely the same thing as the definition using covariance properties under the Euclidean group. (One could take this as the definition of Euclidean space.) However, using an arrow strikes me as a much more accessible definition than the one using transformation groups. And yes, there are many many reliable sources that define a vector this way. (In fact, I just peaked at Misner, Thorne, and Wheeler, and even they adopt this definition of a spatial vector.) Silly rabbit (talk) 16:43, 2 March 2008 (UTC)


 * (my comment here is probably a bit of a non-sequitor) But FWIW, only history and a lot of experimental confirmations have cleared up many of the original mathematical questions and unknowns that initially arose in the wake of the two very successful, very agreeable, yet somewhat historically separated scientific areas: electromagnetics and relativity. For example, mathematical invariance--as has already been pointed out--has been around since at least the 1850's, yet its true usefulness and popularity didn't come to light until the phenomenal success of relativity. On the other hand, as has already been mentioned, even without an upfront and clear application of the condition of invariance, the science of electromagnetics somehow also managed--in the 19th century decades before relativity came into being--to not only accurately predict the physical constant of light, but also to also be one of the few 19th century sciences whose original equations hold true even today under relativistic conditions. Invariance is no doubt a sufficient condition to ensure accurate physical application of entities with magnitude and direction (and this condition can and does do so in, I will be the first to admit, the more graceful manner in terms of expression and mathematics), but a sort of existential quantifier on the history of science nearly proves the existence of an equivalent set of ideas that shouldn't be ruled out altogether (even if not quite as graceful and as well-discussed as invariance). --Firefly322 (talk) 02:59, 3 March 2008 (UTC)

Historical remarks
I think that we should tread very lightly around the history of vectors: e.g. Kreyszig, Differential geometry, University of Toronto Press, 1959, p.9 (reprinted by Dover, 1991), claims that


 * ''The concept of a vector was first used by W. Snellius (1581 – 1626) and L. Euler (1707 – 83).

My understanding is that the contribution of Gibbs (and later, of Heaviside) was to advance vector calculus based on what we'll today call "vector space approach", as opposed to the quaternion approach prevalent at the time. But it's very tangential to the main goal of the article, which is to explain modern point of view on vectors, not to give a genetic introduction following their long and convoluted history. Arcfrk (talk) 03:24, 3 March 2008 (UTC)
 * I agree about treading very lightly around the history of vectors. My understanding, and hope this isn't too much to share, is that several ancient Greek authors wrote about the parallelogram of velocities in two dimensions (including the unknown Greecian author of Mechanica, Archimedes, and Hero of Alexandria). And yes more than a few details would probably exceed the still incipiently fleshed-out goals of the article. --Firefly322 (talk)  05:18, 3 March 2008 (UTC)

Vectors as now taught follows Vector Analysis (Gibbs/Wilson) as a dumbing down of the quaternion studies in hypercomplex numbers. Evidently the mathematical abstraction needed to be reigned in for clear application to physical science. Shedding the multiplicative structure of algebras leads to linear algebra, a field we need to improve in WP. The history of linear algebra is caught up in its applications, the driving force. It is surprising how Euclidean-inspired intuition continues to dominate linear algebra education. Fortunately the WP editors have helped move understanding along by the department-free nature of our colaboration. For the history of Vectors then, we have the early period noted by Firefly322 and others, and then the explosion of the 1890's which produced a Quaternion Society (1899 - 1913) and a conservative reaction curtailing studies to three-dimensions.Rgdboer (talk) 20:41, 6 March 2008 (UTC)
 * I wouldn't define a group as a ring without multiplicative structure; nor a set as a list without order. Generally we want to build upwards from the simple, specific, and concrete, towards the complex, general, and abstract; so to me defining vectors from quaternions would be backwards (if not unhistorical). Pete St.John (talk) 21:14, 6 March 2008 (UTC)
 * Putting aside the historical perspective on vectors, doesn't axiomatic math start from the rigorous, the simple, the general, and the abstract and move towards the specific and the concrete? Perhaps the approach of physical science is what is being mentioned here?   --Firefly322 (talk) 11:47, 7 March 2008 (UTC)
 * Indeed it's not so simple as I made it out. "Simple, specific, concrete" may clash; what's simple may be general, etc. However, in axiomatic systems, defintions get more complex with development, as they have more referents (lemmas); so we define Ring (with two operations) after we define Group (with one). So there is a tendency to increasing complexity. But we also define matrix (as an array of numbers) before we define linear transformation, the former being simpler, but also less general; that is, simple and specific and concrete (a technique for solving systems of simultaneous equations) and get more complex, general, and abstract later (inner product spaces). But there are whorls in the current. Pete St.John (talk) 17:51, 7 March 2008 (UTC)

norms
Is the main distinction between "spatial vectors" and "vectors", that between normed vector spaces and vector spaces? I think part of my own confusion was the sense from the lead that we were talking about vector spaces in general, when I surmised from the discussion that we were not. So perhaps:


 * Spatial Vectors are used in the sciences to represent magnitudes together with directions, such as momentum or velocity, and comprise normed vector spaces in theoretical mathematics.

But perhaps their are more requirements, e.g. a euclidean inner-product space? I can accept the popular motivation of "magnitude and direction" in the definition, but if there is reference to the actual mathematical object that defines the subject, it would be unambiguous as well as pedagical. There is no harm in mentioning links to the general from an article on the particular, and we can keep at the undergraduate level. Pete St.John (talk) 19:41, 3 March 2008 (UTC)

Vectors and pseudovectors
In my opinion, this article, and many other articles I looked at, conflates the notion of vector and pseudovector way too much to be understandable to a general audience (or a pure math audience for that matter). More precisely, I feel like the mathematical notion of vector and the physics notion of vector are being conflated. Really they just happen to have the same name. It is true that attempts at distinguishing these notions occur in this article, but more needs to be done. Much of the confusing occurs from the fact that every time pseudovector or cross product is used, the reader is sent to that article, so that these articles combine to bounce the reader around without ever properly addressing the issue.

As a side note, the notion of vector and vector field are also confused (in fact the term vector field never arises in this article, though many examples given are of vector fields). RobHar (talk) 16:26, 31 July 2008 (UTC)


 * I think the last section addresses the issue of vectors versus pseudovectors, no? So I don't really understand the problem.  I mean, the article *could* start off by talking about general covariance under coordinate transformations, but I hardly think that this would be understandable to a general audience (or even a general mathematical audience).  To address your final point, where exactly do you believe that vector fields are discussed?  I don't see anything which would qualify as a vector field in the article, although perhaps the penultimate section could do with a bit of clarification.  siℓℓy rabbit  ( talk ) 18:58, 31 July 2008 (UTC)


 * Regarding the being bounced around between articles: this article says "the cross product of two vectors is a vector". Then it says, well really if you don't pick an orientation, "the cross product of two vectors is a pseudovector". Then the pseudovector article says a pseudovector is a "quantity" (which is just a pretty word for "thing"). It goes on to say "A common way of constructing a pseudovector p is by taking the cross product of two vectors a and b". This is already somewhat circular. Furthermore, which version of the cross-product is being used is not clear. If I take (1,0,0)×(0,1,0), as someone who has likely recently learned about cross-products at school, I get p:=(0,0,1), so from what I've read, (0,0,1) is a pseudovector. But it's not: if I apply that reflection matrix to p I do indeed get -p (obviously what's happening is that the cross-product has values in Λ2R3, where R3 denotes not really Euclidean three-space, but rather the standard representation of O(3), and Λ2R3 is the second exterior power of the standard representation, and this is sort of where wiki leads us next...). The pseudovector article goes on to say that really they "are more naturally analyzed as bivectors". The bivectors section says they are "an element of the antisymmetric tensor product of a tangent space with itself" (which is related to my vector field comment, I certainly think that the P-vector article has no qualms in confusing vector with vector field, and tensor with tensor field). This statement is probably cryptic at best for most anyone who knows what a vector and a cross-product is. It also says it's something in Geometric algebra (which I've never heard of), and talking about Clifford algebras is probably not the best way to convey the meaning of what is going on. From all this, it's still not clear where a pseudovector lives, though bivector's antisymmetric tensor product of a tangent is hinting at it (for those who know math). If we go over to the cross-product article, things are made slightly clearer, but not without mentioning the hodge star.


 * As for the "Vectors, pseudovectors, and transformations" section: this section says: "However, other vectors are defined in such a way that, upon flipping through a mirror, the vector flips in the same way, but also acquires a negative sign. These are called pseudovectors (or axial vectors), and most commonly occur as cross products of true vectors." So pseudovectors are vectors, but they are also not. This section does explain the example of angular momentum, but there's probably a lot of people that come to this article not knowing what angular momentum is. And looking that up would just bring you back to a cross-product and more circular definitions.


 * Regarding my comment about vector fields: the examples of electric and magnetic fields, for one, though perhaps the article really only means the electric and magnetic field at a specific point, but that's certainly not clear. The penultimate section certainly is completely inspired by the notion of vector field. And in the last section, there's no need to say something like "In the language of differential geometry". Perhaps I'm being too critical on this point, but it just seems as though parts of the article are written at least from the viewpoint of vector fields, and this seems to cloud what's actually going on.


 * My point is that I think the article could be more carefully written so as to be less muddled, and thus more clear. RobHar (talk) 22:31, 31 July 2008 (UTC)


 * Regarding vectors and vector fields: In math, I think, you often say a vector is an element of the tangent space at a point in a manifold. In this construction, all vectors always have "base points". In physics, where the manifold is almost always R3, the tangent spaces are all trivially identifiable with one another, and people almost always talk about vectors without base points. When a physicist talks about a vector with a base point, he's almost always talking about a part of a vector field. I imagine that this is what RobHar was getting at. In other words, in the physics presentation, a vector is an element of R^3 (with certain transformation properties, etc.), and the vector from (0,0,0) to (1,1,1) is the same vector as the vector from (1,1,1) to (2,2,2). I spent some time pondering this issue a while back, and couldn't think of a way to present this that was better than the article as it is now. But there is an untapped potential to present the topic in physics language and to differentiate the math and physics usages in various places.


 * In fact, there's arguably a case for making two articles, "Intro to spatial vectors for mathematicians" and "Intro to spatial vectors for physicists and engineers". I haven't thought this through too carefully, nor would I want to put in the time to execute this split, but maybe someone else feels this is a worthy cause? --Steve (talk) 22:36, 31 July 2008 (UTC)
 * Not sure if you've looked through the archives, Steve, but such an effort has been considered. Since then, I've kept my eyes out for references to support such a split.  Here are two reliable references that would support this sort of new article:
 * Vectors and tensors in engineering and physics (1997) By Donald A. Danielson {its preface flat out states that no real reconcilliation between mathematical and engineering concepts of a vector is possible}
 * G.J.Whitrow's reveiw of Vectorial Mechanics
 * I've also looked at the philosophical differences: the physics/engineering idea of a vector tends towards one in the scientific realism/critical realism realm while the mathematical idea of a vector tends towards one in the realm of bourbaki/logical positivism. My way of looking at this current article is as a "geomtric mean" between these two concepts. And judging from Vectors in three-dimensional space-- where J.S.R. Chisholm shows what's involved in combining these two concepts--I think that such a conceptual "geomtric mean" of the two vector concepts can really only be for research math/science groups of professors/graduate students/Ph.D.'s/post-docs.  --Firefly322 (talk) 01:33, 1 August 2008 (UTC)


 * I think I would rather see an expansion of this article than any dramatic structural changes such as a fork. There is clearly room for expansion, and as suggested above in the discussions with Steven Johnson, the text by Arfken and Weber seems to provide a particularly clear presentation of both relevant points of view on the subject.  It might be helpful in planning any rewrite attempts.  siℓℓy rabbit  ( talk ) 00:40, 1 August 2008 (UTC)


 * On a basic level, the difference I'm talking about is between the notion of spatial vector as an element of R3 (or similar) and the notion of a vector in a representation of O(3). These are two different notions, they aren't adequately differentiated in this article. I don't think explaining it like that would be useful, but perhaps saying something like when physicists think of vectors they don't just think of them as elements of R3 (or whatever), but also as things that can be acted on by O(3) (or whatever symmetry group is involved), so that the notion of vector becomes more complicated and splits into pseudovector as well. I'm not sure there should be two articles out of these two concepts, but I think it should be made clear that they are two different concepts, and efforts should be made to make things such as the notion of cross-product in both cases clear (even if only with a parenthetical remark). RobHar (talk) 01:52, 1 August 2008 (UTC)
 * I went to UCSD about 3 months ago and spoke with a physics professor about this article on vectors and ideas I had for it. I told him what I was thinking and showed him what I was reading. Then asked if I was confused or possibly crazy. He said I wasn't. In fact, he said my ideas were sound.  Anyway, pseudo-vector and ideas of symmetry came up, but differences between R3 and 0(3) didn't.  I'm familiar with R3, but 0(3) is not something I've ever seen in book on vectors.  Do you have a reference explaining what R3 and 0(3) mean relation to each other? Preferably on wikipedia or on a website.  --Firefly322 (talk) 02:08, 1 August 2008 (UTC)
 * Some context: O(3) is the orthogonal group. It consists of all spatial rotations as well as improper rotations (in 3d, these are always spatial reflections).  The notation R3 refers to 3-dimensional real coordinate space.  The notion of vector qua real coordinate quantity is taken up in the article tuple. siℓℓy rabbit  ( talk ) 02:31, 1 August 2008 (UTC)


 * (reply to RobHar): The article says that a vector can be represented as an element of R3, but that it is a geometrical object in Euclidean space.  At some point, I started a draft where this was made a little more explicit, but I abandoned it since I didn't feel it was very suitable for inclusion.  In the shortrun, I believe that the article currently over-emphasizes the former coordinate-dependent approach over the latter coordinate-free one.  In keeping with the idea of a vector as a geometrical figure (rather than a tuple), it would be best to present vector addition, scalar multiplication, and the dot and cross products first in geometrical terms, and then afterwards in terms of the Cartesian coordinates.  This is how things are usually done by, for example, Gibbs and Wilson.  Most of the recent run of calculus books also adopt this approach.


 * I also think that the fact that vectors are acted upon by various, possibly different, structure groups is dealt with in the appropriate place in the article: after the "standard" material has been introduced. siℓℓy rabbit  ( talk ) 02:52, 1 August 2008 (UTC)

New title proposal
As already pointed out by others at the beginning of this section, "vector (spatial)" is a somewhat questionable (or "suboptimal") title for this article, which is supposed to have a restricted scope. Any vector in a vector space, even the most abstract vector space, is a spatial vector (or space-vector) by definition! However, it is difficult to find out a better title, because actually in the literature the subject of this article is simply referred to as a "vector", without further specification.

This is how the many meanings of the term "vector" are classified in the disambiguation page (where I described the special kind of space-vector discussed here as an element of an Euclidean space):


 * {| class="wikitable" border="1"


 * Vector may refer to:
 * Vector may refer to:

Mathematics and physics


 * An element of an Euclidean vector space (or Euclidean-space vector), that is an entity endowed with both length and direction, typically represented as an arrow.
 * An element of any vector space (or space-vector), often represented as a coordinate vector, not necessarily characterized by a length and direction.
 * An ordered list of numbers: a tuple.
 * [...]

Computer science
 * [...]

Biology
 * [...]

Games and fiction
 * [...]

Company names
 * [...]

Other uses
 * [...]


 * }

I insist that the title should be changed, because too generic to indicate the restricted scope of the article. I propose one of the following: By the way, Steven G. Johnson (17:14, 1 March 2008) proposed also However, this title is ambiguos. The many readers who don't happen to imagine the existance of non-normed vector spaces will not be able to guess that this title is supposed to mean "vectors with magnitude and direction". They will think that the article is about the "magnitude and direction of a vector". Paolo.dL (talk) 20:56, 20 August 2008 (UTC)
 * Vector (Euclidean)
 * Vector (in Euclidean space)
 * Vector (elementary physics)
 * Vector (geometry)
 * Vector (geometric), proposed by Silly Rabbit (18:37, 1 March 2008)
 * Vector (mangnitude and direction)


 * For what it's worth, either vector (Euclidean) or vector (geometric) seem the best to me. siℓℓy rabbit  ( talk ) 10:43, 25 August 2008 (UTC)

New title
We have several times written in this page that the title "Vector (spatial)" was sub-optimal, for an article about Euclidean vectors. Indeed, any element of a vector space (even a non-Euclidean space), is a spatial vector by definition! As you can see, taking into account your suggestions, I changed the title to "Vector (geometric)". This is also the title of the corresponding article in "Springer Encyclopedia of Mathematics", and was endorsed by Silly rabbit. I also completed the terminological note at the beginning of the "Overview" section. Paolo.dL (talk) 18:13, 3 September 2008 (UTC)

I believe that, for the above mentioned reason, the old page Vector (spatial) should be redirected to Vector space, rather than to this page (after substituting with Vector (geometry) all the links pointing to Vector (spatial)) Let me know if you agree. Paolo.dL (talk) 18:30, 3 September 2008 (UTC)


 * I disagree with changing the redirect. Who would type the exact letters "Vector (spatial)" into Wikipedia's article-finder? People would type "Vector" instead. I believe that the only people going to the article Vector (spatial) are people clicking hyperlinks from emails or websites outside wikipedia, and those people might as well be sent to the article they intended to find. --Steve (talk) 19:59, 3 September 2008 (UTC)


 * Good point. siℓℓy rabbit ( talk ) 20:07, 3 September 2008 (UTC)


 * Yes, good point. I won't change the redirection from Vector (spatial). However, there are also two pages called Spatial vector and Spacial vector, which redirect here (and I just created a page for Space-vector). I believe it is advisable to redirect all of them to Vector space. Let me know if you agree. Paolo.dL (talk) 20:27, 3 September 2008 (UTC)

Erm, sorry to add a spanner, but per DAB, shouldn't the title be either Vector (geometry) or Geometric vector rather than Vector (geometric)? Geometry guy 17:41, 4 September 2008 (UTC)


 * I'd support Vector (geometry) for page title there are quite a few other articles with similar names, which I think is the least confusing one. --Salix alba (talk) 22:31, 4 September 2008 (UTC)

Any vector is a vector in some kind of space. I don't like "geometric" as much because it doesn't have the automatic association with that level of abstraction.--Loodog (talk) 20:13, 5 September 2008 (UTC)


 * Vector (geometry) and Vector (physics) are sub-optimal, in my opinion. These vectors are not used only in geometry, and in advanced physics non-Euclidean spaces are considered. If you wanted to indicate the field, in this case you should be more specific:
 * Vector (elementary mathematics and physics) would be specific enough, but it is too long, in my opinion.
 * Geometric vector is used only by a few authors. Others use Spatial vector. As far as I know, there's even someone who uses Physical vector.
 * Notice that I posted my question on the 20th of August. Only Silly rabbit answered. Then I changed the title, according to Silly rabbit suggestion (supported also by a reference to "Springer Encyclopedia of Mathematics"), and spent hours to replace hundreds of Wikilinks to the old page with updated Wikilinks to the new page.
 * Paolo.dL (talk) 20:35, 5 September 2008 (UTC)


 * If this article is to change title yet again, I think the term "Euclidean vector" may be the best. One can define a "geometric vector" in some other geometry, say the geometry based on affine transformations.  In that case, a vector can still be described as a directed line segment (an arrow), but it now no longer makes sense to talk about length as a scalar value (although "magnitude and direction" would still exist in the sense that some vectors are "commensurable").  Parts of the "Overview" section had been written with this more general view in mind, but I see that they have been eroded over time due to a lack of general agreement about the subject of the article.  I therefore put forward that the article is about (and should be about) vectors in a Euclidean space. siℓℓy rabbit  ( talk ) 01:16, 6 September 2008 (UTC)

Usage of the expression "spatial vector"
I don't agree with the statement that whenever the term "spatial vector" is used, it means, by definition, an element of a vector space. First of all, I challenge you to find any author using the term in this way. Second, I have provided a reference in the article to an author who uses the term spatial vector to refer to a vector in a Euclidean space (that is, a magnitude+direction vector). A quick look at google books shows that this term does have some currency, although mostly in applied areas such as continuum mechanics. <font color="#c00000">siℓℓy rabbit ( <font color="#c00000">talk ) 10:52, 5 September 2008 (UTC)


 * Seconded. All three should continue to redirect to here.  Jheald (talk) 13:13, 5 September 2008 (UTC)
 * Thirded: "Spatial" here is referring to space in the naive sense, i.e., what a 19th century mathematician would have called "ordinary space". It isn't referring to Hilbert space any more than it is referring to topological space. Geometry guy 17:58, 5 September 2008 (UTC)

From Talk:Vector (geometric): "The term spatial vector does not seem right. Its usage seems to occur in the specialist fields of ECG analysis and  soliton waves.  Feynman talks of a space-vector which is plainer English but begs the question of what sort of space we are talking about - the general reader might suppose this is Outer space." [...] Colonel Warden (talk) 08:22, 1 March 2008 (UTC)


 * Silly rabbit provided a reference where the expression spatial vector is used with a restricted meaning, to refer to a "geometric" vector. Geometry guy seems to maintain that this is the correct interpretation. However, if Feynman "begs the question of what sort of space we are talking about" (see previous comment), others (me included) are even more likely to beg the same question. Before Hilbert, the expression "spatial vector" wouldn't be so ambiguous. Now, too many people are aware of "extra-ordinary" spaces. Thus, for those who don't have Geometry guy's point of view or background about terminology (and as far as I understand Feynman has a different point of view), the expression is ambiguous. However, I agree with Silly rabbit that it should be included at the end of the list of possible names for a "geometric" or "geometrical" vector. My point is just that it would be wise not to adopt this ambiguous terminology in the text of Wikipedia articles. Paolo.dL (talk) 20:09, 5 September 2008 (UTC)


 * Silly rabbit, I browsed the reference you provided, and I am not sure that it supports your point. The author (John H. Heinbockel) seems to use the word "spatial" to distinguish two representations of a physical quantity (Eulerian or spatial and Lagrangian; see quotation below). So, he is not trying to distinguish a vector in an Euclidean space from a vector in a more abstract vector space. (Notice, also, that the expression "space vector" is not contained in the analytic index of his book). Paolo.dL (talk) 20:09, 5 September 2008 (UTC)


 * Let xi denote the initial position of a material particle in a continuum. Assume that at a later time the particle has moved to another point whose coordinates are xi. Both sets of coordinates are referred to the same coordinate system. When the final position can be expressed as a function of the initial position and time we can write xi = xi(x1, x2, x3, t). Whenever the changes of any physical quantity is represented in terms of its initial position and time, the representation is referred to as a Lagrangian or material representation of the quantity. This can be thought of as a transformation of the coordinates. When the Jacobian J(x//x) of this transformation is different from zero, the above set of equations have a unique inverse xi = xi(x1, x2, x3, t), where the position of the particle is now expressed in terms of its instantaneous position and time. Such a representation is referred to as an Eulerian or spatial description of the motion. Heinbockel, part 9, 2001, page 228.


 * He discusses spatial vectors elsewhere, and in that context "spatial" means "from the ambient Euclidean space" (see Geometry guy's point below). The distinction between the two equations of motion seem to be unrelated to the notion of a spatial vector.  See Heinbockel, part 6, 2001, pages 141-145.  <font color="#c00000">siℓℓy rabbit  ( <font color="#c00000">talk ) 22:05, 5 September 2008 (UTC)


 * You are right, he distingushes between a spatial vector (Cartesian coordinates in Euclidean space) and a surface vector (curvilinear coordinates in a Riemannian space). Not a wise choice of terminology, in my opinion, but this is just my opinion. Feynman and Colonel Warden seem to support my opinion, but Heinbockel seems to be quite an authoritative author as well. Thank you. Paolo.dL (talk) 22:56, 5 September 2008 (UTC)

(ec) I didn't think I would have to explain what I meant by naive: "my" view is the view of the general public, for whom "space" almost always refers to a straightforward, 3 dimensional, Euclidean sense. It has absolutely nothing to do with my background. My background simply allowed me to elucidate this point with reference to 19th century terminology. An encyclopedia does not build its terminology around Feynman's point of view, or mine, or anyone elses: it makes itself as accessible as possible to as broad an audience as possible. Geometry guy 21:59, 5 September 2008 (UTC)


 * We are discussing here about terminology to distinguish a vector in an Euclidean space from a vector in any vector space. If you are naive, you don't need this distinction. Whoever needs this distinction knows about abstract vector spaces (for instance, Heinbockel, 2001, pages 141-145 needs to distinguish between an Euclidean and a Riemannian space). You need an adjective to identify an ordinary vector space if and only if you are aware of extra-ordinary spaces. Thus, in this context (which is not a naive context) the adjective "spatial" is just the worst choice, in my opinion. If you are naive, and you only know the ambient Euclidean space, you just use the word "vector", without specifications, and without even considering the existance of other kinds of vectors. My point is closer to the view of the general public than yours. I am more naive than you! :-) Paolo.dL (talk) 22:26, 5 September 2008 (UTC)


 * I think you are mainly referring to the page move. That's fine: Vector (spatial) was not a good choice, and Vector (geometric) is an improvement, although both conflict with naming guidelines. However, your extension of this idea to a redefinition of the word "spatial" to mean "in a vector space" as opposed to "in (ordinary) space" is a bridge too far. Step back a bit, and reconsider your views in the light of the comments they have generated. Please remember, your opinion, my opinion, anyone's opinion, do not matter so much as what reliable secondary sources have to say. The smiley is appreciated, because we should not be arguing over who is closer/better/wiser/more naive. Seek compromise, not division. Geometry guy 22:46, 5 September 2008 (UTC)


 * As I already wrote: "My point is just that it would be wise not to adopt this ambiguous terminology in the text of Wikipedia articles." (20:09, 5 September 2008). And of course, in the title of this article. It was not my intention to divide. I only wanted to prove that the expression "spatial vector" is ambiguous, and I am still convinced that there's no fault in my logic. When I wrote that I am more naive than you, I did that as a rethorical conclusion of a detailed explanation, meant to be a respectful answer to a comment of yours starting with this sentence: "...my view is the view of the general public". I seek agreement, not division (nor compromise). I would love to be perceived as a friend, rather than an enemy. Paolo.dL (talk) 23:38, 5 September 2008 (UTC)


 * Don't worry, I don't want to redefine the expression. I accepted Silly rabbit's point. See my edit. Paolo.dL (talk) 00:11, 6 September 2008 (UTC)
 * Okay, that's good, thanks. We are more or less on the same page now. Geometry guy 17:00, 6 September 2008 (UTC)

Just my 0.02 €, but when reading "spatial vector" I would think that "spatial" means "pertaining to space as in Space" rather than as in mathematical space, indeed because the latter would be totally pointless as any vector is in a space. Indeed Spatial redirects to Space, which is specifically about physical space. -- A r m y 1 9 8 7 ! ! ! 18:17, 6 September 2008 (UTC)


 * Well, your point is worth attention. Actually, Heinbockel calls "spatial vector" a vector in Euclidean 3-D space. Possibly, he would not call spatial vector a vector in Euclidean 2-D space, or n-space (n>3). Silly rabbit, I think we actually don't know what is the exact meaning given by Heinbockel to this expression, and it is plausible that this meaning does not coincide with the definition given in this article to the word "vector" (=Euclidean vector). Indeed, it seems that there are two plausible interpretations of the adjective "spatial":
 * naive (classical physics): "spatial" (from space) is more specific than the adjective "Euclidean", or
 * non-naive (mathematics): "spatial" (from vector space) is less specific than the adjective "Euclidean".
 * A third interpretation (spatial = Euclidean) is possible but less plausible. If this is true, then "spatial vector" might not be used as a synonymous to "geometric vector" and "Euclidean vector", contrary to what we maintained in the article. Paolo.dL (talk) 01:20, 7 September 2008 (UTC)


 * (Remember the paradox which creates ambiguity: the "naive" interpretation is only used in contexts where the word "space" is used non-naively, to indicate all kinds of vector spaces; authors of "naive" books, such as introductory physics books, where the word "space" is used "naively", do not need to use the expression "spatial vector"!) Paolo.dL (talk) 01:37, 7 September 2008 (UTC)


 * I don't think this is true. The whole point of the word "spatial" (in its basic sense, which is mathematical) is to indicate three dimensionality. It has nothing to do with vector space, topological space or Hilbert space.
 * As you point out, its meaning is clearest in a naive context, in which case its role is to distinguish vectors in space from vectors in the plane: the latter, clearly, are not "spatial". Higher dimensions are a more recent innovation and used to be referred to as hyperspace, but this has largely fallen out of fashion except in science fiction. Geometry guy 09:27, 7 September 2008 (UTC)


 * The second interpretation is completely implausible: who would bother to write "spatial vector" rather than just "vector" if the former phrase isn't going to be any more specific (or different at all) than the latter? -- A r m y 1 9 8 7 ! ! ! 09:54, 7 September 2008 (UTC)


 * So, we do have a "spatial space", right? :-) (where spatial = 3D Euclidean, and space = vector space in general) But this is only the naive+non-naive (schizoid) interpretation. The other is completely non-naive. I insist that both interpretations are plausible. You fail to notice that:
 * Vector spaces are spaces, thus authors are free to use the word "spatial" to refer to any vector space (not only n-dimensional, but also non-Euclidean vector spaces). This is perfectly plausible.
 * There exist other kinds of vector, that do not belong to a vector space, and do not refer to a basis set (see vector), for instance tuples used in programming languages (1D arrays) or in statistics (probability vector). Or "biological" vectors. Those are not "spatial" at all.
 * Paolo.dL (talk) 17:09, 7 September 2008 (UTC)


 * "You fail to notice that" is not the way to engage in a sensible discussion. I have noticed these points. A topological space is also a space. A space is also a typographical object and a key on a keyboard. So what? Geometry guy 18:33, 7 September 2008 (UTC)

Geometry guy, it was not my intention to offend you! You wrote that the adjective "spatial" "has nothing to do with vector space". Army1987 wrote that "my second interpretation is completely implausible". My previous comment was an answer to these two statements. You might maintain that, for some authors it is customary to use the adjective "spatial" to refer only to a particular kind of space. In this case, I would agree, because this is what Silly rabbit proved. You might even be able to prove that this is true for many (rather than just some) authors. Notwithstanding this, by a general and widely used linguistic convention which most readers perfectly know and accept, "spatial" is an adjective meaning "related to space", (as well as geometrical means "related to geometry"). In a book where the word "space" is used to indicate both an Euclidean and a Riemannian space, readers who are used to apply this linguistical convention are likely to assume, initially, that the word "spatial" has something to do with both vector spaces. If you decide, in this context, to use "spatial" to indicate only the 3D Euclidean space, this is a legitimate but linguistically inconsistent choice, almost as inconsistent as stating that the Rienmannian space is a "non-spatial" space. Silly rabbit proved that this is the choice made by Heinbockel, part 6, 2001. However, I find it hard to believe that all authors are willing to accept this terminology. The best authors minimize the usage of ambiguous terminology. That's why I suggest not to adopt this terminology in WIkipedia.

In sum, my point is that, contrary to what you seemed to maintain, the word "spatial" has two meanings: restricted and non-restricted. I also explained that in my opinion it is linguistically inconsistent to give, in the same text, a non-restricted meaning to the word "space" and a restricted meaning to the related adjective "spatial". Almost as inconsistent as stating that "vector spaces are not spaces". Paolo.dL (talk) 15:56, 13 September 2008 (UTC)


 * There would be no point in using "spatial vector" to mean "a vector in a vector space" because all vectors (in the mathematical sense) are members of a vector space, by definition. It'd be like saying "wet water". To show how ridiculous would that be, consider Minkowski space (the vector space, not the affine space). Consider a standard basis of it. It has four vectors. One of them has a squared norm which has a different sign than the other three. Consider that vector. Consider the span of that vector. It is a vector space, of course. You might want to call its elements "spatial vectors" — and they form the thing that a particular inertial observer calls time. Weird, huh? --<span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; white-space: nowrap;">A r m y 1 9 8 7 ! ! ! 16:09, 14 September 2008 (UTC)
 * (As for "It would be inconsistent to say that vector spaces are not spaces", remember that not all skew fields are fields, and that not all multivalued functions are functions, though that's not exactly my point.) <span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; white-space: nowrap;">A r m y 1 9 8 7 ! ! ! 16:16, 14 September 2008 (UTC)


 * You seem to misinterpret my previous comments. I perfectly agree that, in a mathematical context, using "spatial vector" to mean "vector in a vector space" is riduculous. My point was completely different. I showed that, in a mathematical context, "spatial vector" is linguistically ambiguous when used to mean "vector in a 3-D Euclidean vector space". In a more general context, however, where some vectors are not members of a vector space, it is neither ridiculous nor ambiguous (see my comment dated 17:09, 7 September 2008).
 * If you wished to maintain that not all vector spaces are spaces, as well as not all multivalued functions are functions, that would be quite an interesting point. However, the definition of "function", in mathematics, is by no means considered to be a "naive" definition. On the contrary, as Geometry guy pointed out, the definition of "space" as an ordinary 3-D space is "naive" in contemporary mathematics. In other words, in a mathematical context, "function" has a more restricted meaning than "multivalued function", while "space" has a less restricted meaning than "ordinary space". Thus, in a mathematical context, all vector spaces are spaces.
 * Notice that the main thread of this comment is that it is important to consider the context where the terms "spatial" and "space" are used. Paolo.dL (talk) 19:44, 14 September 2008 (UTC)


 * I was kinda sure I would be misunderstood − hence the "that's not exactly my point". I didn't mean that not all vector spaces are spaces in the mathematical sense, only that arguments about linguistical consistency are moot. (What is light cannot be dark; all feathers are light; therefore, no feather is dark.) But note that the mathematical sense of space is not the only one, and that doesn't necessarily mean that the other meanings are naive. Compare with field: in mathematics it has a different meaning than in physics, but that doesn't make the latter meaning naive. Likewise, in physics the unqualified term space usually refers to physical space unless otherwise specified, but saying that this meaning is naive sounds a little far-fetched to me. And not just pure mathematicians use (Gibbsian) vectors. So one can use spatial vector to have a less generic sense that any vector in a vector space without being "schizoid".
 * BTW, have you ever heard/read anyone using spatial vector to mean "a vector in a vector space, as opposed to e.g. a probability vector"?
 * (As for vectors in computing, I foresee that this will sound a little like Chewbacca defense, but, for example, if I understand correctly, in recent Fortran versions you can add two vectors together, or multiply a vector by a scalar, as you would do with elements in a real coordinate space. In older versions you had to add/multiply them componentwise using a loop. But this doesn't affect what they are or what they serve for; this shows how irrelevant the issue of whether they are vectors in the mathematical sense is.)
 * Also note that I don't think that physical space is a space in the mathematical sense. A mathematical space is a set, and in modern mathematics all members of sets are required to be sets themselves, but I'm still trying to figure out which set is the point which was exactly one metre above my center of mass at 04:00 (Central European Time) of 13 September 2008... <span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; white-space: nowrap;">A r m y 1 9 8 7 ! ! ! 20:38, 14 September 2008 (UTC)


 * Good. At least we agree that all vector spaces are spaces!
 * About your example: What is light cannot be dark; all feathers are light; therefore, no feather is dark. This sentence is inconsistent, and it is not moot to notice the inconsistency. I only suggested to minimize this kind of sentences in Wikipedia articles, unless the article is about a subject like "linguistic ambiguity". That's all. Please feel free not to accept this suggestion.
 * Never heard/read anyone using spatial vector to mean "a vector in a vector space".
 * The discussion about naive or not naive is irrelevant. On that, we perfectly agree. Replace "naive" with "invalid in that context". As I wrote, my comment was mainly about the importance of the context.
 * I agree that a physical space is not a space in the mathematical sense. My point is simple: the word space has different meanings in different contexts. Paolo.dL (talk) 22:58, 14 September 2008 (UTC)
 * That sentence about feathers sounds incredibly like "Spatial means 'of space'; all vectors in mathematics are in a space; therefore all vectors in mathematics are spatial." It is wrong, because the two occurrences of space have different meanings. You've taken back the claim that the meaning it has in the former one is "naive", but now you claim it is "invalid in that context". Why? It is just because vectors are a mathematical thing and so only mathematical terminology can be used with them? If so, the usage of field as in vector field is "invalid in that context", as it refers to fields in the physical sense and not to fields in the mathematical sense. Right? <span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; white-space: nowrap;">A r m y 1 9 8 7 ! ! ! 23:25, 16 September 2008 (UTC)

Usage of expression "physical vector"
There was an article called Vector (physical), now deleted because it was almost a duplication of this article. As far a s you know, is there some author who uses the expression "physical vector" to indicate a vector as defined in this article? Paolo.dL (talk) 20:09, 5 September 2008 (UTC)


 * I think the answer is actually no. (I created the vector (physical)  article.) Can't completely go along with the idea that it was almost a duplication. Certainly it wasn't my intention. For example, philosophical differences exist such that the articles physical constant and mathematical constant aren't considered almost duplications. At any rate, the concept I was attempting to describe through the phrase physical vector has sometimes been referred to as a, which turns up almost the same hits as .  Find usage of Gibbsian vector analysis along with references in the article A History of Vector Analysis (DISCLOSURE: I also created around the same time).  A free vector as defined by Gerald James Whitrow in his review of Vectorial Mechanics (DISCLOSURE: another article I created around the same time) is another important aspect of what I had wanted to capture by starting the article vector (physical) (note that Whitrow also uses the phrase Gibbsian vector ).  Another perspective close to what I thought would benefit wikipedia readers can be found in pages 58-81 of George Polya's Mathematical Methods in Science, especially his idea of a set of vectors acting either successively or simultaneously.  But here Polya only uses the word vector. --Firefly322 (talk) 08:30, 13 September 2008 (UTC)


 * Thank you for sharing this interesting information. Possibly, this interpretation of "physical vector" was rejected because "Gibbsian" vectors are 3-D vectors, and advanced physics also uses other kinds of vectors. Paolo.dL (talk) 16:14, 13 September 2008 (UTC)

Everybody just calls it "vector"
As far as I am concerned, I had never heard about physical, geometric, spatial or space- vectors before reading Wikipedia. I use vectors to represent physical quantities, but I call them just vectors, I don't need to distinguish them from other kinds of vectors. Gibbs and Heaviside used the term "vector", without specifications. Hamilton introduced it without specification. I believe the need to distinguish is not frequent. Typically, you just call it vector, and the exact meaning is identified by the context. Paolo.dL (talk) 20:09, 5 September 2008 (UTC)


 * This page is in the unique position of having to describe vectors in all contexts. If the need to distinguish is not frequent, this is one of those infrequent occasions. 71.72.235.91 (talk) 20:45, 22 October 2009 (UTC)

New title (continued)
So why not just call it Vector (mathematics), we basically just need to disambiguate between this an Vector (biology).--Salix alba (talk) 09:41, 6 September 2008 (UTC)


 * Because in pure mathematics, the concept of vector is much broader. See Vector space. This article is specifically about 1st-order tensors, and/or the vectors which act on the Euclidean affine space. Though, Vector (physics) could be a decent title for this article, maybe? -- A r m y 1 9 8 7 ! ! ! 11:56, 6 September 2008 (UTC)


 * Vector (physics) has been suggested and ultimately rejected by editors on both sides of the physics/mathematics divide, although this is obviously better than vector (mathematics). I think maybe my post above about "Euclidean vector" as a possible title may have been missed, and deserves consideration.  Although I would prefer an article structure that keeps the Euclidean properties and affine properties distinct, I wouldn't know how to explain such an article structure in the overview in a way that would meet WP:MTAA.  I think that a possible solution to my own dillemma would be to bite the bullet and rename the article "Euclidean vector".  At least then there is no ambiguity about the intended subject of the article. <font color="#c00000">siℓℓy rabbit  ( <font color="#c00000">talk ) 12:10, 6 September 2008 (UTC)


 * Given the current content of the article, an honest title might be Vector (applied mathematics), so dedicated is it to the Heaviside/Gibbs treatment of vectors in three dimensional Euclidean space. For the title Vector (geometry) a little more ambition is needed: start with the familiar and readily accessible, but go on to suggest generalizations: four vectors, contravariant vectors, covariant vectors, vectors in Hilbert space, vectors in affine space. Even the title Euclidean vector suggests arbitrary dimension; it also rules out four vectors.
 * I think one of the reasons we have difficultly naming this article is that we don't have an introduction to the concept of a vector which covers all points of view. Geometry guy 17:00, 6 September 2008 (UTC)


 * But not only the vectors acting on the three-dimensional Euclidean affine space used in classical physics to model physical space are used in applied mathematics. For example, the signals as in Signal processing live in (typically but not always infinite-dimensional) vector spaces over R (or C), and PageRank uses a vector in the real coordinate space Rn (where n is the number of pages) which is an eigenvector of a particular matrix, and whose i-th component is the PageRank of the i-th page. -- A r m y 1 9 8 7 ! ! ! 18:54, 6 September 2008 (UTC)

What about vector (classical physics)? It explicitly rules out 4-vectors and the like. Note that the title Vector (geometry) doesn't give a clue about the fact that we are dealing with 3D vectors... A r m y 1 9 8 7 ! ! ! 18:20, 6 September 2008 (UTC)


 * Geometry guy raises an interesting point. Maybe someone should propose an introduction on talk which covers all relevant points of view as to the notion of a geometric vector.  Much of the material in the article is not specific to three-dimensional Euclidean space, so requiring the space to be three-dimensional is unduly restrictive.  <font color="#c00000">siℓℓy rabbit  ( <font color="#c00000">talk ) 20:51, 6 September 2008 (UTC)


 * Lets have a look at what we've got on the elementary geometry of vectors. In the plane, we have plane (mathematics), which mostly deals with the spatial geometry of planes (!), and Euclidean plane which redirects to Euclidean geometry: the latter, disappointingly, is entirely about Euclidean plane geometry (dab page linked). There is no article of the form planar vector, as far as I can see. With this in mind we can reevaluate what we have on spatial Euclidean geometry: spatial geometry is a redirect to Euclidean space, which in turn is a treatment of arbitrary dimensions, with no specific discussion of the three dimensional case. Apart from this article, that is pretty much it. Does anyone know of any other major articles on this topic?
 * In short, our coverage is inadequate, and this article has too much work to do in order to patch up the cracks. Giving it a different name is not going to solve the problem unless that name change facilitates a restructuring of this and related articles. Geometry guy 09:53, 7 September 2008 (UTC)
 * The stuff about cross products, triple products, polar vs. axial vectors, and uses in physics and engineering is specific to 3D vectors. But anyway, I think a good idea would be: adding short sections about four-vectors, covariance and contravariance of vectors, and maybe even tensors (each one with a main link followed by one or two paragraphs), and then move the article to Vector (physics). What do you think? -- A r m y 1 9 8 7 ! ! ! 10:09, 7 September 2008 (UTC)
 * I share Silly rabbit's lack of enthusiasm for Vector (physics). This is a mathematical topic even if it is primarily used in applied mathematics, physics and engineering. However, you are beginning to sketch out an article on the general conception of a vector. We need both an article like this and an article specifically on spatial (3D) vector geometry. Geometry guy 11:07, 7 September 2008 (UTC)
 * But if we remove all the things specific to 3D and physics/engineering from this article, it becomes essentially a duplicate of Euclidean space (despite the unfortunate title, that article is mainly about Euclidean vector spaces rather than affine spaces, but that's another matter...) -- A r m y 1 9 8 7 ! ! ! 12:08, 7 September 2008 (UTC)
 * This is not what I am advocating. It is probably best to view the current text as the basis for an article on spatial (3D) vector geometry, then think about what to put in a separate article on all elementary notions of vector. I agree, Euclidean space is confused, and we don't want to follow that at all. Four vectors are not Euclidean anyway. Geometry guy 12:52, 7 September 2008 (UTC)
 * What do you mean by "all elementary notions of vector" exactly? -- A r m y 1 9 8 7 ! ! ! 16:06, 7 September 2008 (UTC)
 * All those notions that engage our "magnitude+direction" intuition. I'm not going to draw a definite line: that is what consensus is for. Geometry guy 16:17, 7 September 2008 (UTC)
 * Huh? There are many kinds of normed vector spaces, including, for example, the space of functions from [0, 1] to R with continuous first derivative, with the norm ||f|| = supx &isin; [0, 1] |f(x)| + supx &isin; [0, 1] |f&prime;(x)|. Any vector in any such space has a norm, and dividing it by its norm you obtain a unit vector representing its "direction". Anyway, the article Magnitude (mathematics) only furtherly confused my ideas about whether "norm" and "magnitude" are synonymous as far as vector spaces are concerned, and the only link in Direction (disambiguation) which could be relevant (i.e. the first one) is close to useless. -- A r m y 1 9 8 7 ! ! ! 17:04, 7 September 2008 (UTC)

(←) Use your intelligence, not what you think I mean, nor what some of Wikipedia's poorer articles say. Would it help if I said "physical intuition"? Perhaps not, so why not use your own intuition and come to your own conclusions. Geometry guy 18:31, 7 September 2008 (UTC)


 * One can be perfectly precise here, without reference to intuition. One generalization of the concept of a vector with magnitude and direction is, indeed, any normed vector space (or, more commonly, any Hilbert space if you want angles and completeness).  However, there is a subset of this general abstract notion, which formalises "direction" in a narrower way, as a precise relationship between the vector and spatial coordinates (or geometry, or whatever you want to call it).  This can be defined as saying that the vector's components have to transform in the same way (contravariantly) with the spatial coordinates under rotations, or it can be expressed more abstractly in differential geometry via tangent spaces.  See many moderately advanced physics texts, e.g. Arfken & Weber, Mathematical Methods in Physics, for this definition, which makes precise the intuitive notion of "direction" relative to space.   Because this is the formalization of the elementary and historical notion of vectors as direction/magnitude in space, or "arrows", it makes sense to have an Wikipedia article that does three things: summarizes the elementary notion of vectors in 3-space at a high-school level, explains that this intuitive concept of "direction" is made precise at an advanced level via contravariance, and gives a pointer to other abstract generalizations in the form of vector (mathematics), inner-product space, etcetera.


 * I feel like a persistent problem in this article is that many mathematically sophisticated editors seem to think that the only way the vector concept can be formalized and made precise is by the general concept of abstract vector spaces over fields, without realizing that there is a narrower, but totally standard, formalization that is more specific to space-related "directions". Both of these formal concepts, the broader and the narrower, are commonly used in physics, engineering, and other fields of applied mathematics, and both deserve articles. —Steven G. Johnson (talk) 19:17, 7 September 2008 (UTC)
 * (interposting) An extraordinarily elegant proposal. --Firefly322 (talk) 08:49, 13 September 2008 (UTC)


 * Yes, I understand what you say about contravariance. (Feynman in  Six Easy Pieces Six Not So Easy Pieces  clearly explains that point without getting too sophisticated with mathematical formalism or terminology — he doesn't even use the word "contravariance".)
 * Now, we have an article Vector space which is about vectors in mathematics in general, with Vector (mathematics) redirecting there. This is completely right and no-one would seriously object to that. Then we have this article, and no-one seems to have clear ideas about (1) what its scope should it exactly be, and (2) how it should be titled. What would you suggest, Steven? -- A r m y 1 9 8 7 ! ! ! 20:18, 7 September 2008 (UTC)


 * (e/c) Could I add my support for the use of the text by Arfken and Weber. Although I don't have it in front of me at the moment, I did examine it at one point and found it to be a very nice model for what we should try to do with this article.  Anyway, I think I more or less completely agree with you on that the intended scope of this article should be this sort of vector, and not the kind of vector which derives its existence from a vector space, or its length from a norm. <font color="#c00000">siℓℓy rabbit  ( <font color="#c00000">talk ) 20:21, 7 September 2008 (UTC)

I'm increasing thinking that what we really need is an article on Vectors in three dimensions to serve as a) a good introduction for the layman and b) to group the results specific to that dimension. Vector (mathematics) could then be a more general purpose article summarising 2D and 3D as well as the other generalisations. Physical uses of vectors could then either be included in one of these articles or perhaphs in its own one Vector (physics). --Salix alba (talk) 21:26, 7 September 2008 (UTC)


 * (interposting) For Vectors in three dimensions, John Stephen Roy Chisholm's book Vectors in three-dimensional space shows us what such an article would be like. Such an approach sounds great, but the results, based on available work and references, are, as Chishom shows, just so-so.  I believe User:Stevenj's proposal is better, simply based on what several authors have tried and results. --Firefly322 (talk) 09:00, 13 September 2008 (UTC)


 * The only logical thing for Vector (mathematics) is to redirect to Vector space as it currently does. In mathematics, the most bizzare objects you can imagine are referred to as vectors, as long as it makes sense to add two of them together and to multiply one of them by a scalar in a field in a way which satisfies some obvious properties. -- A r m y 1 9 8 7 ! ! ! 22:58, 7 September 2008 (UTC)


 * I would recommend against Vectors in three dimensions as the title. The dimensionality of the vector space is not especially interesting here.  Historically, vectors started out as "directions and magnitudes", and as the subject developed and became more precise it seems to have gone in two directions: general abstract vector spaces (and Banach spaces, Hilbert spaces, etc.), and contravariance.  The former are covered in vector space, and the latter (which formalizes the intuitive notion of "having a direction" in space, "arrows," etcetera) seems to be the natural subject matter here.  More specifically, this article should be about contravariant vectors in Euclidean space (as opposed to curved manifolds, spacetime, etc.).  Unfortunately, "contravariant vector in Euclidean space" or "vector (contravariant)" (or "polar vector" or a few other synonyms) is way too sophisticated for most readers&mdash;when most readers (and authors) encounter this subject, it is simply called a "vector" and defined (19th-century fashion) as having a "direction and magnitude".  There isn't any standard elementary-level name that I'm aware of, so we just have to pick something reasonably descriptive and familiar, and then make sure that the article carefully explains its subject matter.  "Vector (spatial)" or "Vector (geometric)" both seem fine to me&mdash;the point is that we are talking about vectors that have some relationship to space/geometry.


 * I'm wary of "Vector (Euclidean)", although I'm not dead-set against it, simply because it's not the Euclidean nature of the vector space that's important here, but rather the Euclidean nature of the space the vectors are contravariant with.


 * More important than the title is how we organize the article to explain the subject matter. Because most readers will have only encountered this subject at a high-school or early-undergraduate level, it's important to review the subject at that level, and be clear that what we are talking about is what is informally and familiarly defined as something "having a magnitude and direction" such as velocity or displacement.  And most of the article should review the usual vector operations and applications at that level (as we do now).  At the same time, the introduction should also say that there are both a more precise definition (contravariance) of "having a direction" (not "any vector in three dimensions" and not any vector with a Euclidean norm) and other abstract generalizations in the form of normed vector spaces etc.  Then there should be one section (called "formal definition" or similar and making it clear that this is for advanced readers) briefly reviewing contravariance at the simplest level and making it clear that this is considered (by most advanced applied-mathematics texts that I know of) as the precise version "having a direction" (I would recommend the Arfken & Weber approach of using transformation of coordinates under rotations, rather than something more impenetrable like tangent spaces), and give examples of things that are and aren't contravariant.  Then point to Covariance and contravariance of vectors for more detail and generalization to other manifolds (although that article currently needs a lot of work&mdash;it's currently so focused on technical manipulations that you can't see the forest for the trees).   Right now, the mention of contravariance is buried under "generalizations"&mdash;that doesn't seem right, as contravariance is better described as a formal definition of "magnitude and direction" vectors (versus other vector spaces) rather than a "generalization" per se. —Steven G. Johnson (talk) 18:07, 13 September 2008 (UTC)


 * You "recommend against Vectors in three dimensions as the title", and I agree with you. Then you recommend for "Vector (spatial)" or "Vector (geometric)". However, in a previous subsection (Usage of the expression "spatial vector") we agreed that "spatial" means "in a three dimensional ordinary space" (at least according to the references provided by Silly rabbit)... Paolo.dL (talk) 09:29, 14 September 2008 (UTC)

An article titled Gibbsian vector analysis would point directly to this school of thought's 19th century origins. In fact, one engineering text book author who has used this terminology makes a very similiar or even the same conceptual distinction to that of Steven G. Johnson's: Vectors and Tensors in Engineering and Physics, by D.A. Danielson, Addison-Wesley, 1992 (2nd edition 2003) (page 16): "Mathematicians often use the word "vector" in a more general sense than in this book, as an element of linear vector space. In that sense, n-tuples (sequences of n numbers) and n x m matrices are also vectors. The directed line segments (arrows) used here are sometimes called Gibbsian vectors to distinguish them from the more abstract vectors." Moreover, Prof. Danielson is not alone among engineering text book authors. Prof. Ismo V. Lindell's Differential Forms in Electromagnetics Published by Wiley-IEEE, 2004 ISBN 0-4716-4801-9 uses the full phrase Gibbsian vector analysis (as the google book preview link shows).

The mathematical physicist Gerald James Whitrow also uses the term Gibbsian vector in his review Mathematical Gazette 1949 of Vectorial Mechanics. Concerning a Vectors in three dimensions title, Gibbsian vector analysis has never been limited to just the three dimensions represented by the x, y, and z unit vectors. It's been known since Willard Gibbs's time that the dot product and cross product holds when one or more additional unit vectors (say unit w for example) is added. --Firefly322 (talk) 11:25, 14 September 2008 (UTC)


 * The cross product is only defined in 3D and in 7D (unless you want it to be something different than a vector, that is). What did you actually mean? --<span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; white-space: nowrap;">A r m y 1 9 8 7 ! ! ! 13:25, 14 September 2008 (UTC)
 * You're absolutely right of course. I clearly need to check some historical texts to see again exactly what was or was not discussed (I have actually read up on this before).  For now I believe, the discussion back then would have been about higher dimensions of unit vectors holding under commutative and associative properties of addition and that 3D cross-products had anticommutativity properties.  --Firefly322 (talk) 13:55, 14 September 2008 (UTC)
 * (Actually, even in 3d the cross product is something different from a vector, if one wants to be precise about contravariance. —Steven G. Johnson (talk) 15:38, 14 September 2008 (UTC))
 * (interposting) I realize that commutative and associative laws may seem pendantic, but in the 19th century few mathematicians or physicists appreciated them and it has been shown that such a lack of understanding is in part what thwarted an appreciation of Hermann Grassmann's work among others. --Firefly322 (talk) 19:49, 14 September 2008 (UTC)
 * (If we want to get that pedantic, I meant that you cannot have a cross product taking two tensors of order 1 and returning a (pseudo)tensor of order 1, and having properties analogue to that of the 3D cross product, in dimension other than 3 — or, if you relax some of the restrictions, other than 3 or 7. It would have to return a (pseudo)tensor of higher order, or a (pseudo)scalar in dimension 2. <span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; white-space: nowrap;">A r m y 1 9 8 7 ! ! ! 16:24, 14 September 2008 (UTC))

Historical note: Josiah Willard Gibbs was (also) an engineer. He developed his vector analysis in the late 19th century with a specific applicative purpose: he offered it as a tool for students of physics. His "vector algebra" was appreciated (and is still appreciated) by physicists because of its simplicity, but was strongly criticized by mathematicians because it was tied to a 3-D ordinary space and not completely generalizable to higher dimensions (the dot product was generalizable, the cross product was only valid in 3-D and 7-D). On the contrary, the "Extension theory" introduced about 40 years before by Hermann Grassmann was already equipped with a wedge product of vectors which was not tied to three dimensions. Grassmann's work (together with Hamilton's quaternion algebra) led to Exterior algebra, Clifford algebra, and eventually to modern abstract algebra. Thus, Gibbs's contribution may be perceived as a dead end in the history of mathematics. However, introductory physics courses all over the world are still based on Gibbs's (aka "conventional" or "traditional") vector algebra. By the way, conventional vector algebra is not only Gibbsian. While Gibbs formulated his theory at Yale University (USA), at the same time in England Oliver Heaviside (another engineer) independently developed an almost indentical theory (see vector analysis). Paolo.dL (talk) 18:10, 14 September 2008 (UTC)


 * The history here has been well-studied by dedicated historians and most of what one finds in the typical classroom texts books, according to professional historians of science is quite frankly faulty. For example, there's no question that in Electrical Engineering Electromagnetics is still taught and praticed using what is in fact Josiah Willard Gibbs system (and I'm under the impression that a large portion of physicists use and/or are fluent in it). As for Hermann Grassmann's brilliant work, alas, it has been shown not to have actually influenced the widespread adoption of vector analysis in the early 20th century; this often confused bit of history is analyzed in A History of Vector Analysis (p. 94 states that although the essence of modern vector analysis could always be found in Grassman's work, others discovered these key ideas independently and that "Grassman's ideas exerted little or no influence on the history of vector analysis. The irony is that they could have; the fact is ...that they did not.") Also telling is criticism of J.D. Jackson's Classical Electrodynamics (1962), which has been given as an example of textbook histories that are [with] "slight exageration...a pack of lies". This is found on pages 338-339 of the book  Fields of Force: The Development of a world view from Faraday to Einstein (1974). So what's the source of this Historical note? Can it be named? (For it is well-known that Heaviside and Gibbs constructed the same vector system independently and that they later united to promote its adoption, which their combined efforts succeeded at doing.) --Firefly322 (talk) 19:09, 14 September 2008 (UTC)


 * I agree, but I am puzzled: what you wrote seems to be perfectly consistent with my note. I wrote that conventional vector algebra (independently formulated by Gibbs and Heaviside) is still alive and widely used in physics (I forgot to add that it is also extensively used in engineering). When I wrote that "Gibbs's contribution may be perceived as a dead end in the history of mathematics", I meant that it could not be generalized to n dimensions, and that successive developements in mathematics (by Giuseppe Peano, Clifford, Henri Poincaré, Élie Cartan, Gaston Darboux, Alfred North Whitehead, etc.), that eventually led to abstract algebra, were mainly based on the previous contribution of other authors (Hermann Grassmann and Hamilton). We discussed these bits of history in Talk:Exterior algebra and Talk:Cross product. The references are provided there, and in the history sections of the corresponding articles. Paolo.dL (talk) 21:45, 14 September 2008 (UTC)


 * Since talk pages are supposed to be for discussing the article, what is the relevance of all that? Physicists still use Gibbs's vectors; the fact that mathematicians now prefer more general, abstract concepts is irrelevant to this article. --<span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; white-space: nowrap;">A r m y 1 9 8 7 ! ! ! 22:23, 14 September 2008 (UTC)


 * Hear, hear. MarcusMaximus (talk) 22:40, 14 September 2008 (UTC)


 * I didn't want to discuss about history. My historical note was not perfectly focused, but it contained information relevant to this subsection (about the title of this article): (1) vector analysis is (in part) tied to a 3-D space, (2) but because of its semplicity it is still widely applied in physics and engineering, and (3) Gibbs was not its only father. Point 2 makes vector analysis interesting for Wikiproject:mathematics. Point 3 makes "Gibbsian vector" a title less acceptable than "Vectors in three dimensions". That's it. Sorry for interrupting the discussion flow. Paolo.dL (talk) 23:14, 14 September 2008 (UTC)


 * What did Gibbs and Heaviside call their vectors? MarcusMaximus (talk) 23:33, 14 September 2008 (UTC)
 * Both used the word vector as an adjective and as a noun. Examples:
 * Heaviside used the phrase: "vector or directed quantity" (another instance of same phrase) (1894)
 * Gibbs used the phrase:Vector Analysis (1881) --Firefly322 (talk) 09:44, 15 September 2008 (UTC)

Shall we move this to Vector (directed quantity)? --<span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; white-space: nowrap;">A r m y 1 9 8 7 ! ! ! 10:29, 15 September 2008 (UTC)
 * If this is a proposal, then I will second it: Vector (directed quantity)
 * As an aside, James C. Maxwell also used the phrase a vector, or directed quantity in his defintion of vector in the book A Treatise on Electricity and Magnetism (1892). Though another possibility for such an article name would just be vector quantity, which Maxwell uses at least nine times in the same document. --Firefly322 (talk) 18:19, 23 September 2008 (UTC)

Pages redirected to this article
Here is a list of titles of pages which redirect to this article, which in my opinion is worth of some attention. It is surprisingly long: This list is, on one hand, incomplete (why "bound vector", and not "free vector"?), and on the other hand, contains pages that should probably point elsewhere ("force vector", "vector theory"). It also contains "relative vector", and there's nothing about relative vectors in this article (what are them, by the way? Is it something like relative position?). Notice that "Axial vector" is not included because it points to pseudovector. Paolo.dL (talk) 13:07, 4 September 2008 (UTC)
 * Bound vector
 * Component (vector)
 * Force vector
 * Physical vector
 * Polar and axial vectors
 * Polar and Axial vectors
 * Polar vector
 * Polar vectors
 * Relative vector
 * Three-vector
 * True vector
 * Vector (classical mechanics)
 * Vector (geometry)
 * Vector (physics)
 * Vector (spatial) [This is for compatibility with previous title of this article]
 * Vector addition
 * Vector component
 * Vector methods (physics)
 * Vector subtraction
 * Vector sum
 * Vector theory
 * Vectors and Scalars


 * I'd say "Polar and axial vectors", "Polar and Axial vectors", "Polar vector", "Polar vectors", and "True vector" should all point to pseudovector. --Steve (talk) 23:09, 4 September 2008 (UTC)
 * UPDATE: I did this. --Steve (talk) 03:49, 14 September 2008 (UTC)

Vector "components" don't depend on the coordinate system
There was a statement near the beginning of the article that said vector components depend on the coordinate system. This is false. Vector components depend on the vector itself and the direction in which the component is taken. The component in any given direction is independent of the coordinate system used. That is, if I represent the a vector p as


 * $$\mathbf{p} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$$,

where i, j, k are mutually orthogonal unit vectors, the numbers a, b, and c are not the components of the vector. There is a component of the vector in the i direction, for example, which is the vector ai, and in the j direction it is bj, etc., but it has no dependence on the coordinate system, only individually on p and i or on p and j. There is also a component of the vector in any arbitrary direction I choose, such as the n direction, but again, it has no dependence on the coordinate system. It will just be a scalar times n. In other words the definition of the component pn of p in the direction of n is


 * $$\mathbf{p}_\mathbf{n} = \frac{(\mathbf{p} \cdot \mathbf{n})}{||\mathbf{n}||}\mathbf{n}$$.

I believe the term the writer was searching for here would be something more akin to "measure number". In the example, the measure numbers in the i, j, k directions are a, b, and c, respectively. And since most people seem to prefer the column matrix or row matrix representation of a vector, (a, b, c), they fill such a matrix with the measure numbers of the vector but erroneously call a, b, and c the x, y, and z components, and then they come to believe that these "components" depend on the x-y-z coordinate system used. Truly, the measure numbers a, b, and c depend on the basis used. In this case the basis is the set of vectors i, j, k. Another basis u, v, w would give different measure numbers,


 * $$\mathbf{p} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} = \alpha\mathbf{u} + \beta\mathbf{v} + \gamma\mathbf{w} $$,

but the components αu, βv, and γw are what they are, independent of the existence of any coordinate system.

I think this also goes to the heart of a widespread problem with the understanding of vectors, in which people think that vectors are column matrices (or row matrices), and that column matrices are vectors, rather than merely being a mathematical representation of vectors in a specified basis. Sometimes this is accompanied by the belief that a vector itself exists only with a coordinate system, and that a vector always must have a physical location with its tail at "(0,0,0)" of some coordinate system and the measure numbers ("components") specify its "endpoint". MarcusMaximus (talk) 07:39, 8 September 2008 (UTC)
 * At least some people use the word component to refer to what you call measure number (I had never heard that phrase before), e.g. uses the phrase "the x component of v" to refer to the scalar v &middot; i = vx rather than the vector vxi. Others call the former "scalar components" and the latter "vector components"
 * (As for column matrices, they are vectors, even if not the kind of vectors that this article is about...) -- A r m y 1 9 8 7 ! ! ! 08:31, 8 September 2008 (UTC)
 * (As for "vector components depend on the vector itself and the direction in which the component is taken", but the direction in which the components are taken do depend on the coordinate system — do you understand what I mean?) -- A r m y 1 9 8 7 ! ! ! 08:45, 8 September 2008 (UTC)

I'll admit measure number is not a universal term, so "scalar component" might be an approximate synonym that sees wider usage.

I believe I understand what you mean when you say "the direction in which the components are taken do depend on the coordinate system". If we assume we are only talking about Cartesian CSs, I think you are saying that the subset of vector components we are interested in depends on the coordinate system, but not the components themselves.
 * —This is part of a comment by MarcusMaximus which was interrupted by the following: Yes, that's what I meant. A r m y 1 9 8 7 ! ! ! 10:27, 8 September 2008 (UTC)

This is probably close to what the original author was trying to say too, in the sense that i, j, k and u, v, w might be called "coordinate systems". I think it is a fuzzy idea, however. Truly, i, j, k is not accurately called a coordinate system. It is just a set of three orthogonal basis vectors without an origin. I can pick an infinite number of right- or left-handed Cartesian coordinate systems that have x, y, and z axes parallel to i, j, and k, or jki, or kij, in any order, with an origin arbitrarily located, and I would still take the same three "components", and the equation for p above would be unchanged. So the "components" of a vector really depend on the basis vectors that are used to take the components, not on a coordinate system. Not to mention that, depending on what kind of coordinate system I choose (cylindrical, spherical), I can get mathematical representations of a vector that don't have "components" at all.

Therefore, I think it is more accurate and more general to just say that the mathematical representation of a vector depends on the coordinate system. MarcusMaximus (talk) 09:40, 8 September 2008 (UTC)


 * I believe I must have been the original author. I find your version an improvement, despite disagreeing with some of your objections to the original.  <font color="#c00000">siℓℓy rabbit  ( <font color="#c00000">talk ) 11:17, 8 September 2008 (UTC)

Wikipedia says: "A vector becomes a tuple of real numbers, its scalar components." Yes, in the tuple there are real numbers, but: So maybe the best choice for naming the real numbers in the tuple are simply real numbers or coordinates? -- SasQ —Preceding unsigned comment added by Sasq777 (talk • contribs) 07:56, 15 February 2009 (UTC)
 * they're not scalars - scalars are independent of the coordinate system. Example: for a given point in space that has a temperature, the temperature will stay the same whatever coordinate system you choose to describe that point's location. But the numbers in the tuple does depend on the coordinate system! So they're not scalars.
 * they're not components - components are vectors along the basis vectors. Real numbers aren't vectors either.

Unit vector: no magnitude?
The article says "A unit vector is any vector with a length of one; geometrically, it indicates a direction but no magnitude." I think this is getting at the fact that in practice, unit vectors are used for defining a direction only. However, the wording seems imprecise at best, because it implies that a vector which just happens to be of magnitude 1 (but is not necessarily so) has no magnitude at all. Obviously not all magnitude-1 vectors are "unit vectors" in this sense. There must be a better way to say this. MarcusMaximus (talk) 23:03, 14 September 2008 (UTC)
 * A vector that just happens to have magnitude 1 (but is not necessarily so) happens to be a unit vector (but is not necessarily so). The unit vector corresponding to a is a/|a|, thus... (Note that, for this to happen, the vector must be dimensionless...) --<span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; white-space: nowrap;">A r m y 1 9 8 7 ! ! ! 10:33, 15 September 2008 (UTC)
 * This sentence is very poor. For example, as noted by MarcusMaximus, it could be read as saying a unit vector has no magnitude. A better sentence is "A unit vector is any vector with a length of one; normally unit vectors are used simply to indicate direction."
 * I made this replacement. Brews ohare (talk) 19:20, 15 September 2008 (UTC)

Scope of the article
Many names have been proposed, but before deciding the name, we should reach an agreement about the scope of the article. My contribution to this effort is limited by the fact that I am not a mathematician. However, according to previous comments, I can summarize the "environment" where this article should find its (not necessarily empty) "niche". Paolo.dL (talk) 19:55, 15 September 2008 (UTC)

[This list is mainly a summary of contributions from the previous subsection. Please feel free to edit it Paolo.dL (talk) 19:55, 15 September 2008 (UTC)]
 * IN THE PLANE
 * Plane (mathematics), which mostly deals with the spatial geometry of planes
 * Euclidean geometry, disappointingly, is entirely about Euclidean plane geometry. (Euclidean plane, Euclidean plane geometry redirect there; Plane geometry is a disambiguation page linking there)
 * Planar vector is missing
 * 3-D ("SPATIAL") EUCLIDEAN GEOMETRY:
 * Spatial geometry is a redirect to Euclidean space, which in turn is a treatment of arbitrary dimensions, with no specific discussion of the three dimensional case.
 * Vector analysis currently points to this article for the definition of "vector", and disappointingly does not even give a complete list of the vector operations defined by Gibbs and Heaviside, including dot product, cross product, (scalar and vector) triple product.
 * FOUR VECTORS
 * Four-vector, space-time vectors commonly used in physics
 * Minkowski space
 * VECTOR SPACES IN GENERAL
 * Vector (mathematics) redirects to Vector space
 * PROPERTIES WHICH CAN BE USED TO RESTRICT THE SCOPE OF THIS ARTICLE
 * Pseudovector
 * Covariance and contravariance of vectors
 * Inner product space
 * Normed vector space
 * Magnitude (mathematics)
 * Orientation (geometry) (this refers also to rigid objects in 3-D)

No support for move to Vector (directed quantity). The separate retitling discussion below this has not been requested at WP:RM,and can be done by anyone.--Fuhghettaboutit (talk) 18:37, 1 October 2008 (UTC)

Requested move
Vector (geometric) → Vector (directed quantity) — Don't ask me why I am proposing this while I opposed the very similar title Vector (magnitude and direction): I would answer that this one takes six less keystrokes to type... More seriously, this description was used by Gibbs, Heaviside, and Maxwell, and quantity sounds slightly less "abstract" than magnitude to me: I would almost always take quantity to refer to a physical quantity. — <span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; white-space: nowrap;">A r m y 1 9 8 7 ! ! ! 18:57, 23 September 2008 (UTC)

Survey

 * Feel free to state your position on the renaming proposal by beginning a new line in this section with  or  , then sign your comment with  . Since polling is not a substitute for discussion, please explain your reasons, taking into account Wikipedia's naming conventions.


 * Oppose. There's a reason all three of the sources cited are nineteenth century: a quantity can't be directed, and it took much time to break out of this Victorian murk. Let's not take Oliver Heaviside, of all men, as our guide and philosopher; paradox swarms around him like a swarm of mosquitoes. Septentrionalis PMAnderson 03:56, 24 September 2008 (UTC)
 * Oppose. I have to strongly disagree.  Just open up any basic linear algebra or vector calculus book.  The standard term is "geometric vector".  No need to start naming articles based on original articles written centuries ago.  Otherwise we'd be moving "derivative" to "fluxion".  --C S (talk) 04:44, 24 September 2008 (UTC)
 * Oppose. "Geometric" describes how this kind of vector behaves; it is a very good description of purpose, and furthermore the geometry can be used to give a precise definition. "Directed quantity" attempts to be descriptive, but it's unavoidably vague: What kinds of quantities does it refer to, and what does it mean for such a quantity to be directed? It quickly becomes circular unless you fall back on the underlying geometry. I would much rather the article title suggest a correct interpretation of the objects in question. Ozob (talk) 14:51, 24 September 2008 (UTC)
 * Oppose. This is a superfluous move. The subject of vectors as covered in this article relates mainly to geometry, and hence the _(geometric) suffix. I don't see how replacing this with _(directed quantity) makes things any clearer. To the contrary, it seems to me to be replacing a clear, concise suffix with vague weasel words.&mdash;Tetracube (talk) 23:59, 24 September 2008 (UTC)
 * Oppose I'm not 100% happy with current name Vector (geometry), Vector (mathematics) may be better. Vector (directed quantity) just seems to focus on one particular aspect. --Salix (talk): 07:21, 25 September 2008 (UTC)
 * Comment. No, Vector (mathematics) would be totally inappropriate, as the vectors described in this article aren't the only kind of vectors used in mathematics. Vector space is about the general concept of a vector in mathematics, and Vector (mathematics) redirects there, as it should. --<span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; border: thin solid black"> A r m y 1 9 8 7 ! ! !  08:53, 25 September 2008 (UTC)
 * Also note that the current title is Vector (geometric), not Vector (geometry), and, having to choose between these two, I'd prefer the former (unlike the 99% of other cases, in which I would prefer the noun rather than the adjective), because these vectors are seldom studied in pure maths anymore (search for the comment by Paolo.dL starting with "Historical note:" above, and the two following comments). But on the other hand, these are geometric vectors (whatever this means)... --<span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; border: thin solid black"> A r m y 1 9 8 7 ! ! !  14:54, 25 September 2008 (UTC)
 * Oppose. We've been through this already.  <font color="#c00000">siℓℓy rabbit  ( <font color="#c00000">talk ) 14:53, 28 September 2008 (UTC)
 * Oppose. It's probably the most correct location technically, but I think would prove confusing for readers.  After all, "The names of Wikipedia articles should be optimized for readers over editors, and for a general audience over specialists."--Loodog (talk) 16:01, 28 September 2008 (UTC)

Proposal
Rename to Euclidean vector.
 * Support --Salix (talk): 07:18, 28 September 2008 (UTC)
 * Support (per all of the discussion above and below). <font color="#c00000">siℓℓy rabbit  ( <font color="#c00000">talk ) 14:38, 28 September 2008 (UTC)
 * Support because it's a standard term, while it's not clear at all what a "geometric" vector is supposed to be and "directed quantity" is obsolete terminology. --Hans Adler (talk) 15:56, 28 September 2008 (UTC)
 * Support. Not sure this type of vector can only be Euclidean, but I like the accessibility it grants to readers.--Loodog (talk) 20:22, 28 September 2008 (UTC)
 * Support. Standard term with clear scope, and consistent with naming conventions. Notions of vector beyond this scope (such as affine vector and four vector) can be discussed elsewhere and perhaps, ultimately, vector (mathematics) can be transformed from a redirect into an overview. Geometry guy 20:32, 28 September 2008 (UTC)
 * Oppose. If anything, Vector (Euclidean) would be better, as these things are most commonly called just vectors.  In general, I'm concerned that people will get the mistaken impression that it is the dimensionality or Euclidean-ness of the vector space itself that is important, rather than the Euclidean-ness of the manifold the vectors are contravariant with.  As long as the article text is clear, I suppose it will be okay.  —Steven G. Johnson (talk) 05:54, 29 September 2008 (UTC)
 * Please forgive my ignorance, but what does "Euclidean-ness of the manifold the vectors are contravariant with" mean? Geometry guy 21:52, 1 October 2008 (UTC)
 * Which term don't you understand? Contravariance?  Manifold?  Euclidean manifolds vs. non-Euclidean manifolds?  —Steven G. Johnson (talk) 02:14, 2 October 2008 (UTC)
 * I join GeometryGuy in the ignorance here. Few people besides you, Steven, have said anything about Euclidean manifolds and contravariance in this discussion.  I don't mean to downplay the importance of those properties, because it seems that you believe they are vitally important to defining the type of vector described in this article in a mathematical sense.  However, to an audience not trained in that level of mathematics, is there a more transparent way to define what a vector is for the purposes of this article? MarcusMaximus (talk) 02:43, 2 October 2008 (UTC)
 * Yes, and I think it is a major persistent problem here that most editors don't know the formal definition of what the article is (or should be, I think) about. (Basically, contravariance evolved as a formalization of the original intuitive notion of "having a direction", as distinct from the general notion of abstract vector spaces that was developed later.)  It's actually not that complicated, as long as we restrict ourselves here to ordinary Euclidean space and don't worry about curved manifolds.  (The contravariance article in Wikipedia is hopelessly unintelligible.)  I think I should probably write a short introductory-level summary in a section of the Talk page, drafted for inclusion in the article if people agree.   Again, I should emphasize that most of the article should be at the elementary level of "magnitude and direction" and "arrows", but somewhere in the article it should explain precisely what these things mean.  —Steven G. Johnson (talk) 03:30, 2 October 2008 (UTC)
 * I would very much appreciate such a summary. Maybe after we all read it, the fusion of all our minds will be able to come up with a title better than Euclidean Vector. MarcusMaximus (talk) 04:45, 2 October 2008 (UTC)
 * I understand all the individual terms, and know perfectly well what covariant and contravariant vectors are, and how they formalize the concept of a (tangent) (co)vector by how it transforms under changes of coordinates. A vector being "contravariant with" a manifold is new to me. I agree that Wikipedia's coverage of the contravariance approach is rather poor, but it is no more fundamental to what a vector is as definitions via derivations or equivalence classes of curves. Geometry guy 07:53, 2 October 2008 (UTC)
 * PS. I would also welcome an introductory summary. Even if it doesn't turn out to be useful here, Covariance and contravariance of vectors needs help! Geometry guy 08:05, 2 October 2008 (UTC)
 * Let me join in the chorus asking for an explanation. I, too, know what "Euclidean", "contravariant", "manifold", and "vector" all mean individually, but I also don't understand what you're trying to say. Ozob (talk) 15:16, 2 October 2008 (UTC)
 * I don't understand. How can you know what contravariant means, but not know what "contravariant with" means?  Contravariance of a vector space is always defined with respect to a manifold, i.e. with respect to some spatial geometry.  (Literally, the terms "contravariant" and "covariant" mean "varying against" and "varying with", after all.) —Steven G. Johnson (talk) 15:58, 2 October 2008 (UTC)
 * Support per the discussion below. Ozob (talk) 00:39, 2 October 2008 (UTC)

Discussion

 * Any additional comments:


 * The current title (geometric) does not seem to be broad enough, because the vectors described in this section include basically all vectors used in physics and engineering. I would estimate that more than half, if not a supermajority of the vectors used in those fields are not geometric in nature; that is, they are not position vectors or unit vectors used to define axes of rotation.  Is my understanding of the term "geometric" too narrow? MarcusMaximus (talk) 07:09, 27 September 2008 (UTC)
 * I don't want to decide what the term geometric means exactly; but the article needs to be fixed, too. "A vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B [...]". Where is the terminal point of the gravitational field vector at the center of mass of the Moon? The same holds with any vector which doesn't have the dimensions of length. --<span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; border: thin solid black"> A r m y 1 9 8 7 ! ! !  10:04, 27 September 2008 (UTC)
 * Yes, your understanding of geometric in this context is too narrow. —Steven G. Johnson (talk) 00:04, 28 September 2008 (UTC)
 * How about Euclidean vector the title would defines the main mathematical characteristic that they vectors is an euclidean space, which variations on "geometric" do not. Plenty of reference for the use of the term.--Salix (talk): 10:22, 27 September 2008 (UTC)
 * Excellent proposal. I came here to find out what "vector (geometric)" is supposed to be, as I had no idea. "Euclidean vector" would have been immediately clear. --Hans Adler (talk) 13:46, 27 September 2008 (UTC)
 * One minor point against that is that it would focus more on the mathematics of these vectors than on their use in physics and engineering. These vectors aren't very interesting from the pure math POV, as I pointed out above; also, all three-dimensional vector spaces over the real field with a definite-positive inner product are isomorphic with each other, so defining what differentiates physical vectors from ordered triples of real numbers from the POV of pure mathematics would be almost pointless. For the use of Euclidean vectors in pure mathematics we already have Euclidean space; this article is supposed to be about their uses (in particular, that of three-dimensional Euclidean vectors) in physics and engineering. <references group="Note" /> --<span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; border: thin solid black"> A r m y 1 9 8 7 ! ! !  14:40, 27 September 2008 (UTC)
 * BTW, if you do that, please use Vector (Euclidean) rather than Euclidean vector, as most links would simply use "vector" as their text, and the former title allows the pipe trick. --<span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; border: thin solid black"> A r m y 1 9 8 7 ! ! !  16:12, 27 September 2008 (UTC)
 * The purpose of an article title is not to make it easy for editors to use the pipe trick! Vector (Euclidean), like Vector (spatial) and Vector (geometric), go against standard naming conventions. Geometry guy 16:42, 27 September 2008 (UTC)
 * No, it isn't, but that shows that, in the contexts in which these vectors are usually used (namely, physics and engineering), they are usually simply referred to as vectors, not as Euclidean vectors, so using more than "Vector" as the title-without-the-parentheses is not a really good idea. If your objection is about the use of an adjective as a disambiguator, what about Vector (Euclidean geometry)? --<span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; border: thin solid black"> A r m y 1 9 8 7 ! ! !  17:56, 27 September 2008 (UTC)
 * As I've pointed out before, the defining property of the "magnitude-and-direction" vectors like displacement, velocity, force, etcetera, is not that they are in a Euclidean space but that they are contravariant with one. As someone else said, the mere fact that the vectors are three-dimensional and have an L2 norm is not especially interesting; what is interesting, and what does distinguish them from arbitrary abstract inner-product spaces of the same dimensionality, is their relationship to a (Euclidean) manifold.  (See e.g. Arfken & Weber, Mathematical Methods in Physics.)  —Steven G. Johnson (talk) 19:03, 27 September 2008 (UTC)
 * I agree, but which title do you suggest? --<span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF; border: thin solid black"> A r m y 1 9 8 7 ! ! !  19:09, 27 September 2008 (UTC)
 * As I said earlier, I think this obsession with the title is a red herring. The vast majority of sources refer to these things as "vectors" without qualification, and something really technically precise like "vector (contravariant with Euclidean manifold)" will repel most of the intended audience.  We just have to settle for something more-or-less reasonable like "Vector (geometric)" or "Vector (spatial)"...I honestly don't much care as long as it's not directly misleading like "Vector (three-dimensional)" or "Vector (physics)"...and focus instead on making the article text clear.  —Steven G. Johnson (talk) 20:08, 27 September 2008 (UTC)
 * Have "Vector (physics)", "Vector (engineering)", "Vector (physics and engineering)", or "Vector (physical sciences)", "Vector (mechanics)" ever been proposed? MarcusMaximus (talk) 21:11, 27 September 2008 (UTC)
 * Yes, repeatedly. And repeatedly been rejected, on the grounds that many other kinds of vectors are used in physics and engineering.  (e.g. Advanced physics is mostly concerned with infinite-dimensional vector spaces of functions, e.g. wavefunctions in quantum mechanics.  And linear algebra in general is all over the place in engineering, to the point where computer architectures are judged by their ability to run matrix libraries like LINPACK.  Contravariant vector spaces are only one small corner of physics and engineering beyond the freshman level.)  Nor are contravariant vectors restricted to particular subjects like classical mechanics (which also uses many other vector spaces).  Using those names for this article would be positively misleading.  —Steven G. Johnson (talk) 23:54, 27 September 2008 (UTC)
 * I have to plead ignorance of the infinite-dimensional advanced physics topics you mentioned. However, I'm not convinced that the prevalence of linear algebra and matrix math in engineering necessarily changes the answer you would get from an engineer if you asked him to describe a vector as it is used within his field; I think you'd more or less get a description matching the content of this article.  Are you saying this article is really about vectors in contravariant vector spaces? MarcusMaximus (talk) 00:32, 28 September 2008 (UTC)
 * Yes, because contravariance is merely the (standard) formal definition of the old intuitive notion of a vector having a "direction" in space. But this level of formality is too much for most of this article (and too much for the title), since these kinds of vectors are usually introduced at a high-school level; the mention of contravariance should be there, but probably just in one subsection on "formal definition" or something like that.  (As for the opinions of your hypothetical naive engineer, they are irrelevant; the use of many other types of vector spaces in engineering is an indisputable fact.) —Steven G. Johnson (talk) 01:59, 28 September 2008 (UTC)
 * This contravariance nature is what happen in an Euclidean space, so Euclidean vector would nicely define that precisely. I'd actually say that pseudo-vectors are not really vectors at all but they are bivector, its a happy accident that the Hodge dual maps the space of bivector to the set of vectors in 3D. You only get this confusion in 3D as in other dimensions the Hodge dual does not map to vectors.--Salix (talk): 06:59, 28 September 2008 (UTC)
 * I'm not entirely happy with Euclidean vector or (better, as the common name is just vector) vector (Euclidean) as it seems to imply that what matters is that the vector space is Euclidean (note that the tangent space itself is always Euclidean in some sense, even for a curved manifold), rather than the space with which the vector is contravariant being Euclidean. However, I don't have strong feelings about this, as long as the article text is clear.
 * (Yes, pseudovectors and cross products are "accidents" of three dimensions, but that's irrelevant for Wikipedia; the exploitation and naming of this accident is extremely standard, and it is not our place to tell readers that another interpretation is better. Worse, arguing over what mathematical sets "really" are is a route to madness, as the history of mathematics attests (recalling centuries of pointless sophism about negative numbers, irrational numbers, infinitesimals, ...).  Certainly, bivectors are more general, but that generality is not needed in the cases where pseudovector-notation is used, and generations of scientists have found pseudovector notation to be very convenient in three dimensions.  Anyway, I'm not sure why you are bringing this up here.)  —Steven G. Johnson (talk) 17:19, 28 September 2008 (UTC)

Reworked
I've just reworked the article, creating a basic properties section with standard operations, and focusing on 3D Cartesian coordinates, with a consistent notation. Hopefully this will make it a bit more accessible for high school students and such like. I've also factored out the physics material creating a separate section for that.

There still seems to be a lot missing which deserve summary sections --Salix (talk): 20:02, 27 September 2008 (UTC)
 * Rotation and other linear transformations (this would make discussion on pseudo-vector more meaningful)
 * Vector fields
 * More basic physics
 * Discussion on vector spaces
 * History


 * I still think it's fundamentally wrong to list contravariance under generalizations, rather than making it clear that this is actually the standard way to precisely define the original notion of a vector as something with a magnitude and direction, as opposed to the more general subsequent concept of abstract normed vector spaces. (But this is not your fault; it was that way before your edit.) —Steven G. Johnson (talk) 20:13, 27 September 2008 (UTC)

Vectors expressed in different bases
A reader of this article might notice that all of the vector operations described in the basic properties section are performed with vectors expressed in the same basis e1,e2,e3. However, the vast majority of nontrivial real-world problems in physics and engineering involve the use of vectors known in terms of different bases (for example, in my aerospace experience: inertial, Earth-fixed, north-east-down, downrange-crossrange-up, body-fixed, sensor-fixed, IMU-fixed, gyro-fixed, line-of-sight-fixed; not to be confused with reference frames).

Therefore, I think this article needs a section about direction cosines and what are commonly known as transformation matrices or rotation matrices, to demonstrate how a vector can be expressed in different bases but is still the same vector, as in



\mathbf{a} = a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3 = u\mathbf{n}_1 + v\mathbf{n}_2 + w\mathbf{n}_3 $$

and how to get from one basis to another using column matrices to represent vectors in different bases:


 * $$\begin{bmatrix} u \\ v \\ w \\ \end{bmatrix} = \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} $$

where cij is the direction cosine relating ni to ej. If we can refer collectively to e1,e2,e3 as "e" and to n1,n2,n3 as "n", then the matrix containing all the cij is known as the transformation matrix from e to n. Some people call it the rotation matrix because they think of it as the "rotation" of a vector from one basis to another. My personal preference is direction cosine matrix because it contains direction cosines. MarcusMaximus (talk) 23:42, 27 September 2008 (UTC)
 * I did try to address this briefly in the Other_bases section. But you are right we do need something on changing basis, rotations and linear transformations in general. I going to be away for a week so I'll not be able to do this though. --Salix (talk): 23:51, 27 September 2008 (UTC)

I added the section, heavily based on my above post. MarcusMaximus (talk) 07:52, 29 September 2008 (UTC)

Derivative of a vector
This article probably needs a section on the derivative of a vector, since many applications require the derivative. However, my experience with vectors centers around kinematics and dynamics, so I conceptualize the derivative of a vector in terms of reference frames. Is there a more general definition for the derivative of a vector? MarcusMaximus (talk) 06:39, 2 October 2008 (UTC)

I decided to go ahead and add a section on the derivative of a vector. I hope it is sufficient and clear. MarcusMaximus (talk) 08:36, 2 October 2008 (UTC)

Contravariance and the meaning of "having a direction"
Here is a precise definition of what I think the article should be about, and an explanation of this "contravariance" thing I keep yammering about.


 * The earliest definitions of a mathematical "vector," and most elementary treatments today, define it as something having a "magnitude and a direction", something that can be "represented by an arrow." From the standpoint of mathematical precision, however, these definitions leave something to be desired [ editorial note: as evidenced by the continual confusion over what this article is about ].


 * One more precise definition of vector can be found in the general concept of a normed vector space or Banach space, but these concepts are also considerably more general than the original notion of vector. For example, in this definition, a triplet (&rho;1,&rho;2,&rho;3) of the electric charge density at three points in space can be considered a perfectly good abstract vector space, has a perfectly good Euclidean norm given by the mean-square charge density, and is a perfectly good element of R3, even though it is a triplet of scalars that has nothing to do with the intuitive notion of a "direction" in space.  (This is not a completely artificial example: the infinite-dimensional Hilbert space of charge-density functions is an extremely important vector space in functional analysis, because it is central to the study of Poisson's equation as a linear operator.)


 * Instead, there is another way to define vectors (delineating a subset of all abstract vector spaces) which makes precise the notion of "having a direction," and includes classic examples of vectors like velocity, force, and electric field, while excluding triplets of scalars. This definition is the concept of contravariant vector spaces. Although contravariance has many sophisticated formulations and generalizations mentioned below, the simplest form of the concept, in three-dimensional Euclidean space with Cartesian coordinates, is not too complicated and is presented here.


 * The starting point is that when we say that, for example, the electric field "has a direction", we mean that it has a direction in space&mdash;that is, it has some relationship to another vector space, the vector of spatial Cartesian coordinates, that we can precisely define. In particular, let $$\vec{x}=(x_1,x_2,x_3)$$ be a column vector of Cartesian coordinates in space, and let $$\vec{v}=(v_1,v_2,v_3)$$ be a column vector of a contravariant vector space (one "having a direction", to be defined).  The key thing is to look at what happens to these coordinate vectors under rotations.  If we rotate the spatial coordinate system, that corresponds to replacing $$\vec{x}$$ with $$\vec{x}' = R\vec{x}$$, where R is a 3&times;3 rotation matrix (an orthogonal matrix, preserving magnitude in the Euclidean norm).  If $$\vec{v}$$ is contravariant, it means that the same rotation transforms $$\vec{v}$$ to $$\vec{v}' = R\vec{v}$$.  This is, in fact, the simplest definition of contravariance: the components of contravariant vectors transform like the spatial coordinates under rotations.


 * This corresponds to the intuitive notion of a vector as something "having a direction". For example, if you have a force that is pushing you in the +x direction, it might be represented as (+F,0,0), and if you rotate your spatial coordinate system by 90 degrees so that x becomes y, in the new coordinate system you will need to represent the same force as (0,+F,0).  On the other hand, if you have a triplet of charge densities at three points, (&rho;1,&rho;2,&rho;3), the charge density is the same no matter how you orient your coordinate system (we say that the charge density is a scalar: invariant under rotations).  Other examples of contravariant vectors include the standard examples of velocity, momentum, electric field, etc.; an example not drawn from physics and not limited to three dimensions is the gradient vector (field) of any scalar function (technically, the gradient is covariant, but there is no difference between covariance and contravariance in Euclidean space with Cartesian coordinates).


 * As mentioned above, there are many generalizations of contravariance. In differential geometry, it is generalized to non-Euclidean curved manifolds (where one must distinguish between contravariant and covariant vectors), and it can be defined in a coordinate-independent fashion via tangent spaces. Contravariance can also be generalized  to tensors of higher or lower rank (a contravariant vector is a rank-1 tensor). In special relativity, contravariant four-vectors are those that transform like space and time under rotations and boosts (transformations to moving frames of reference).  If one includes not only rotations but also improper rotations (e.g., mirror flips), then the cross product of two contravariant vectors is a pseudovector rather than a vector (gaining an extra sign flip under improper rotations), and pseudovectors themselves are a special representation of a more general notion of bivectors.  In Euclidean space, contravariant vectors are most commonly called simply vectors, or occasionally the term polar vector (nothing to do with polar coordinates) is used to distinguish them from pseudovectors (also called axial vectors).  One can also define transformation properties of contravariant vector fields $$\vec{v}(\vec{x})$$ in Euclidean space, which under a rotation R go to $$\vec{v}'(\vec{x}') = R\vec{v}(R^{-1}\vec{x}')$$.   The outer product of two contravariant vectors is a rank-2 tensor, and in general one can define tensors of any rank r (which can be thought of as r-dimensional $$3\times3\times\cdots\times3$$ matrices that, under rotations R, are multiplied by one factor of R or its transpose for each index).  The basic description of contravariance as defined above can be found in many sources, e.g. a standard reference is Arfken and Weber, Mathematical Methods for Physicists.

Does this clarify matters? Again, most of the article should be written at the elementary "magnitude and direction" and "arrows" level, but I think we need to have one subsection that gives the precise definition of "direction" in the modern sense of contravariance.

—Steven G. Johnson (talk) 16:54, 2 October 2008 (UTC)


 * Thanks, Steven. The simplest reading I can draw from this is that contravariance means that a vector "reacts" to rotations, and this reaction is due to the property we call "direction".  Assuming I am interpreting it right, that seems pretty clear now. MarcusMaximus (talk) 17:20, 2 October 2008 (UTC)


 * Just be careful to remember that the rotations are not "physical" rotations (we're not talking about actually moving any objects), they are just rotations of the coordinate system. Now do you see why I am worried about Euclidean vector?  I'm worried about feeding into the notion that this page is about any element of R3 with a Euclidean norm, which would include things like triplets of scalars that don't actually correspond to the modern understanding (contravariance) of the "magnitude and direction" concept (nor to the intuitive notion, for that matter).  —Steven (spelled with a "v") G. Johnson (talk) 18:01, 2 October 2008 (UTC)


 * I am still somewhat puzzled. Let me see if I can explain in my own words what I think you're saying; I think part of the reason for the difficulty is that you seem to have a physics background and some of the rest of us have a pure math background.


 * You want to distinguish at least three different types of vectors: One type is the kind of vector which is a triplet of numbers (and so represents a point in R3) but has no other meaning; while it's called a "vector" (with no adjectives) the fact that it may be considered as starting from the origin is irrelevant because it's really just a point in space. The second and third kinds are what you're calling "contravariant" and "covariant" vectors. I think what you mean is that these are respectively elements of the tangent space and the cotangent space (to a certain point). In R3 you use the standard metric to identify these two types of vectors, but as you say later these are different on a smooth manifold.


 * I would call a vector in the first sense a "point", not a vector. To me, three-dimensional Euclidean space ought to be considered as a principal homogeneous space (aka torsor) over a point with structure group R3. (Here R3 is considered as a vector space.) The reason for doing so is that the origin is really not important in such circumstances; it's nice when you're doing computations, but nothing intrinsic distinguishes it from any other triple. Whereas in a vector space, the origin is distinguished as the additive identity. And since any two points in a torsor have a well-defined "difference" (which in this case will be vector), you don't lose any of the geometry of the situation.


 * I would call the second and third types simply "vectors" in the present situation. In the case of manifolds, I might distinguish them as "tangent vectors" (your "covariant vectors") and "cotangent vectors" (your "contravariant vectors"), but I'd be more likely to be talking about "vector fields" and "differential forms".


 * If I understand you correctly, you'd like the present article to make an attempt to distinguish the R3 with its standard inner product from R3 considered as a smooth manifold and identified with all its tangent spaces. Is that right? Ozob (talk) 21:23, 2 October 2008 (UTC)


 * I think you may be obscuring the point. Everyone here is talking about vector spaces, not arbitrary triplets of numbers, and when you have a vector space you have an additive identity and hence a meaningful origin.  But there are lots of three-dimensional vector spaces over the reals that are not contravariant.  For example, consider the triplet of charge densities that I gave as an example above.  This is a perfectly reasonable three-dimensional vector space over the reals: because Poisson's equation is linear, it is quite meaningful and useful to talk about adding charge densities or multiplying them by real numbers.  But such vectors are invariant under coordinate rotations of the geometry associated with the charges, and hence they are not contravariant; correspondingly, they do not fall under the original (intuitive) notion of vectors "having a direction".  And, of course, there are plenty of other useful vector spaces in R3 that have nothing to do with geometry at all, e.g. vectors of statistical quantities (e.g. populations) in a three-component Markov process.  All such things are perfectly good abstract vector spaces.  But, in the conventional understanding, the original notion of vectors as having direction in space, what is introduced at an elementary level by the "arrow" representation, corresponds to contravariant vectors only.


 * Yes, you can relate contravariance to tangent spaces in differential geometry, but I think the simple coordinate-based approach is much more useful and accessible here, restricted to Euclidean space. And yes, you're right that with many of these things one is ultimately talking about vector fields, but that is not necessary (and one can equally well talk about the contravariance of the vector field at a fixed point in space).  What is necessary to define contravariance is some notion of geometry (the manifold) in addition to the vector space alone.  Contravariance is not a property of the vector space alone, but rather a property of the vector space in relation to the geometry.  For a vector field, the relevant manifold is obviously the domain of the vector field; for tangent spaces (as defined by e.g. derivations), you are also defining them explicitly with reference to functions on a particular manifold.  For things like the force or the velocity, they are implicitly defined with respect to motion in some geometry.  All these notions get more complicated on curved manifolds, which is why I don't see it as worthwhile to treat that case here.


 * Let me phrase my concern in a more mathematical way: I'm concerned that people will view the subject of this article as any three-dimensional vector spaces over the field of real numbers with an L2 norm, and that is not the conventional understanding of the original "magnitude and direction" vectors in physical applied mathematics and the physical sciences.


 * (All of the definitions I'm using come straight out of standard texts of physical applied mathematics.) —Steven G. Johnson (talk) 23:03, 2 October 2008 (UTC)


 * OK, now I'm definitely confused. I think we agree that abstract vector spaces (and inner product spaces) aren't attached to any manifold, so there's no notion of covariance or contravariance for their elements. (Am I right?) Are you saying that there's some additional data that we can attach to a vector space that makes it behave like a tangent space or cotangent space, but doesn't require the presence of a manifold? Or are you saying that you'd like the article to be about the tangent bundle to R3, that is, vectors with a chosen basepoint? Or are you saying something else entirely? (If it's the last, can you give me give a definition or tell me where I can look it up? I'm really clueless about applied math and physics.) Ozob (talk) 00:51, 3 October 2008 (UTC)


 * I gave the definition, and a standard reference, above already. Contravariant vectors, for a Euclidean manifold, are in a vector space whose components transform like the spatial Cartesian coordinates under rotations.


 * That is, to use your words, there is additional data attached to a (contravariant) vector space saying that it behaves like a tangent space, which does require the presence of a manifold&mdash;contravariance is defined with respect to a manifold. In the common case that we're talking about here, however, the manifold is Euclidean and the choice of a particular basepoint is irrelevant&mdash;the manifold is translationally invariant.  —Steven G. Johnson (talk) 01:03, 3 October 2008 (UTC)


 * Hmm. So you're saying that contravariant vectors are elements of R3 with its standard inner product and the standard action of the orthogonal group O(3)? Ozob (talk) 19:03, 3 October 2008 (UTC)

(←) Maybe I can help Ozob: in pure maths language and in this simplified setting, contravariant vectors are elements of a three dimensional irreducible representation of SO(3) (or equivalently a three dimensional irreducible representation of O(3) on which -I acts by minus the identity). Of course there is only one such representation up to isomorphism and the isomorphisms are unique up to scale, but scale is important in physics because it carries dimension (in the sense of dimensional analysis).

One problem with this definition is the focus specifically on SO(3), the 3 by 3 orthogonal matrices of determinant one. This requires identifying space with R3, which in turn requires choosing a basepoint, and an orthonormal frame (ordered basis). Consequently in this approach one never really says what a "Euclidean/contravariant/whatever" vector is, only what it looks like with respect to a frame, and how it transforms under changes of frame.

A more invariant approach would describe space as an affine space whose group of translations is a 3 dimensional inner product space V. Then contravariant vectors would be elements of three dimensional irreducible representations of SO(V). Again any two such guys are unique up to an isomorphism unique up to scale.

The invariant approach raises the question "what is V?" Will any 3 dimensional inner product space do? Steven doesn't like this because there are three dimensional inner product spaces V which do not represent space, and on which the action of the orthogonal group SO(V) has no physical meaning (what does it mean to rotate the charges at three different points?) Hence he prefers to work with R3 and live with the SO(3) ambiguity in the choice of frame. Any other choice would do, however, as any two 3 dimensional inner product spaces V and W are isomorphic, with the isomorphism unique up to the action of O(V) and O(W), and the sign ambiguity is the topic of another discussion!

Basically, one's approach to vectors depends upon which ambiguities one is prepared to live with, and which philosophical distinctions one cares most about. Wikipedia, however, does not have an opinion. Geometry guy 20:11, 3 October 2008 (UTC)


 * Actually, as far as I understand it, there is an addition nuance to Steven's version of things. In his view there seems to be an additional Euclidean space X representing the physical world on which O(3) (or SO(3)) acts.  The collection F of all orthonormal frames on X gives a particular principal homogeneous space for O(3).  A vector is then an O(3)-equivariant map from F to R3 corresponding to taking a measurement with respect to a frame.  This point of view is obviously important for physics, where one may not be able to identify the "space" in which the vector lives, but can certainly identify (and use) the way it transforms.


 * That said, every three-dimensional representation of O(3) can be realized in this way (up to isomorphism, or isomorphism up to isomorphism, or something), so it really makes no difference if we think of Euclidean space as being R3 equipped with an action of certain matrices, or some other model. It seems better to treat vectors in a purely axiomatic way (as directed line segments in a Euclidean space), as long as one is willing to accept that a "Euclidean space" makes sense.  All of the basic operations on vectors can then be constructed in a purely synthetic fashion without any dependence on the model of Euclidean space under consideration.  <font color="#c00000">siℓℓy rabbit  ( <font color="#c00000">talk ) 20:43, 3 October 2008 (UTC)


 * My understand is: contravariant vectors are elements of the tangent space to R3 which is naturally isomorphic to R3 × R3 (i.e. free vectors with general base points). Properties of the tangent space follow naturally from those of the base space. That is the isomophism is operation preserving for +,-, scaler multiplication, dot product and linear transformations (multiplication by matricies). Its a handy convenience to restrict ourselves to vectors with the origin as base point and be sloppy about distinguishing the tangent space at zero T0(R3) from the base space R3.
 * Covariant vectors arise as the Wedge product product, $$\wedge$$, maps pairs of elements of T0(R3), to the space of 2-vectors, $$T_0({\mathbf R}^3)\otimes T_0({\mathbf R}^3)$$. The Hodge star operator, *, gives an isomorphism of this space to T0(R3). The cross product is really composition
 * $$* \circ \wedge: T_0({\mathbf R}^3) \times T_0({\mathbf R}^3) \to T_0({\mathbf R}^3)\underset{\wedge}{\otimes} T_0({\mathbf R}^3) \ \underset{*}{\approx}\ T_0({\mathbf R}^3)$$.
 * As the wedge product is skew symetric the cross product does not commute with linear transformations. --Salix (talk): 20:46, 3 October 2008 (UTC)


 * Ah, but that's not the meaning Steven is using above! I would like to thank Geometry Guy and Silly Rabbit for their explanations. I think I get it, and I have to say: Wow! That's a great idea!


 * If I were to sum it up in a few words I think I would say that a contravariant vector is one that knows about the reference frame. Let me try to give a more detailed description: Denote space (the usual physical space we live in) by X. Inside space I choose a reference frame: I fix an origin and three orthonormal basis vectors. Call this reference frame F. I am happy with my reference frame, but Sauron is not, so he chooses another reference frame G.


 * The One Ring is located somewhere in X, and the Fellowship is currently carrying it towards Mount Doom. Sauron and I both know the Ring's location; with respect to F the Ring has position (x1, x2, x3), and with respect to G it has a different position (y1, y2, y3). The ring's position does not depend on which reference frame we choose, so after accounting for differences in some way, the two sets of coordinates should be the same. There are two possible differences between our two reference frames: One is that we may have chosen different origins (wherever I am versus Mount Doom), and the second is that we may have chosen different basis vectors (East, North, and Up versus something else, like Northwest, Northeast, and Down). The first we can harmonize by translation, and the second we can harmonize by proper rotation; that is, if I transform my reference frame into Sauron's, then the coordinates of the Ring transform in the same way, and after transformation we see the same set of coordinates for the Ring. Therefore position is a (rather trivial) example of a covariant vector.


 * But position is a covariant vector essentially by definition. A more interesting example is velocity. Velocity does not change under translation, but under rotation it transforms in the same way that position does. For instance, suppose that both Sauron and know how fast the Fellowship is moving. In my reference frame F, their velocity is (v1, v2, v3), and in Sauron's reference frame G, their velocity is (w1, w2, w3). If I want to know what Sauron sees in his reference frame, then all I need to do is rotate myself so that I am pointing the same direction as he is; the velocity I observe transforms according to the rotation I make, so velocity is also an example of a covariant vector.


 * On the other hand, there are plenty of three-dimensional vectors which are not covariant. For example, one can form a vector whose first coordinate is the temperature, second coordinate is the barometric pressure, and third coordinate is the time until second breakfast. (Ignore for the moment the problems with this; in particular the problem that it's always time for second breakfast.) The coordinates of this vector are the same for Sauron and me even though we have chosen different reference frames. If I rotate myself so that I am pointing in the same direction as Sauron, this vector does not change, and therefore it is not covariant.


 * I'm sure there are subtleties about the definition that I haven't considered, but I think that's the general idea. Ozob (talk) 00:26, 4 October 2008 (UTC)


 * I think we're on the same page. The only quibble I would make is that you are using "covariant" when you should be using "contravariant" (yes, I know the names are confusing; I don't know the etymology).  For Euclidean space they are equivalent, though.  —Steven G. Johnson (talk)

I think I agree with Silly rabbit's first paragraph, but not the second paragraph. If you focus only on the basic operations on vectors (addition, multiplication by scalar, norms, etcetera), then you are drifting back to the viewpoint of a "Euclidean vector" being any three-dimensional vector space over the reals with an L2 norm. The definition of a contravariant vector has to exclude things like a triplet of values of a scalar field (e.g. the charge density) at three points in space (which is a perfectly good Banach space, but is not contravariant).

Also, while I appreciate the pure-math preference for coordinate-independent definitions, without exception they require a level of mathematics that isn't appropriate in this article except as a brief mention and a pointer to other articles. The standard way to introduce this topic (at a more formal level than "magnitude and direction" or "arrows"), accessible to undergraduates, and adopted in numerous textbooks, is to define a contravariant vector in a coordinate-based approach: requiring the components to transform like the Cartesian coordinates of "space" (what rabbit calls X) under rotations. (As Geometry guy put it, defining a contravariant vector by "what it looks like with respect to a frame, and how it transforms under changes of frame," relative to changes of frame in the spatial coordinates.)

NPOV does not require that we take no position on what this article is about. We should have different articles on different notions of vector spaces, from general abstract vector spaces to the original notion of "magnitude and direction" vectors in the physical sciences. Every standard reference that I'm aware of, at least those that give a formal definition beyond the "arrow" level, defines the latter as contravariant vectors in Euclidean space (with many later generalizations thereof). And since these things are taught at a very elementary level starting in high school, there needs to be an elementary-level article on the topic, giving the simplest version of a precise definition and pointers to more sophisticated treatments and other generalizations.

—Steven G. Johnson (talk) 23:47, 3 October 2008 (UTC)


 * We can of course take a position on what this article is about, and we need a better article on contravariant vectors, not just a redirect to a confusing one. However, if this article is about the elementary concept of a vector as an arrow in space, then it needs to have no preference as to how that idea is best formalized. Don't forget that Euclidean geometry is a perfectly elementary topic that has been taught to school children for millenia, while the cartesian approach is relatively recent. The view of Euclidean spaces as L2 spaces is likewise rather modern, and the description of charge distributions using Euclidean spaces demonstrates the power of the abstraction: the charges are perfectly contravariant, but just with respect to a different space. The misinterpretation of this as physical space and physical rotation is a flaw in the interpretation, not the mathematics. No physicist would make that mistake, and few others would even be aware of it. But the unitary group of a Hilbert space has multiple vital physical interpretations, beyond the original geometrical inspiration. Geometry guy 21:01, 4 October 2008 (UTC)ctor spaces represent "directions".


 * Charge densities are not at all contravariant. Contravariance does not mean that you can change bases within the vector space (by this definition, every vector space would be contravariant!), but that a length-preserving change of basis (rotations) within the vector space mirrors that of another space (manifold) representing position.  The fact that you seem to think that contravariance is about unitary operations in general indicates to me that you are fundamentally misunderstanding the concept.


 * The fact that Cartesian approaches are more recent than the geometry of Euclid is irrelevant; vectors are much more recent than both, and in any case we are talking about the concept as it is presented today, not 2000 years ago.


 * Every standard text that I'm aware of formalizes "magnitude and direction" "arrow in space" vectors as contravariant vectors with respect to Euclidean manifolds. Do you have a good reference to support the notion that other vector spaces, e.g. abstract Banach spaces, are presented as a formal definition of the original "arrows in space" "magnitude-and-direction" vectors, rather than a generalization of this concept (quite a different cup of tea)?  (Nor do I suggest that this article be a replacement for the article on contravariance in general.  Contravariance has been generalized far beyond Cartesian descriptions of Euclidean manifolds in three dimensions, and those generalizations should be described elsewhere.)


 * —Steven G. Johnson (talk) 21:14, 4 October 2008 (UTC)


 * I'm misunderstanding nothing. What is the other space representing position? Position in what? It could just as easily be position in a space of charges as position in a 10-manifold of string theory. Contravariance is not well defined until you decide what the manifold of positions is, and you can make whatever choice you like, as long as you then fix it. Some choices are more physical than others, but it isn't the fault of the mathematics if you make a bad choice.
 * Euclidean geometry is as relevant today as it was 2000 years ago. Inner product spaces are quite different from abstract vector spaces and Banach spaces and you have no reason to bring the latter into this discussion. Inner product spaces carry much of the geometry of Euclidean spaces, and we can use or misuse this as we wish. Geometry guy 21:56, 4 October 2008 (UTC)


 * Give one reference that uses "contravariance" in the way you describe. By that definition, every vector space is contravariant (with itself).  Why would anyone define a special term if it is true for all vector spaces?


 * In practice, the concept of contravariance is used where there is a clear notion of a "spatial" manifold, and other vector spaces that are contravariant with that. This is certainly true of the original concept of vectors, as distinct from later generalizations, which was used for physical concepts like velocity and electric field that are contravariant with respect to the physical (approximately) Euclidean position space.


 * Abstract inner-product spaces are a generalization of "magnitude and direction" "arrows-in-space" vectors, not a formal definition of the original concept and the concept as it is still taught in high-school and freshman physics/math classes. Name one reference to the contrary.  I've already named an authoritative reference that supports my explanations.


 * —Steven G. Johnson (talk) 22:57, 4 October 2008 (UTC)
 * Why indeed? Good question. Now try reading as well as writing, instead of framing the discussion in your terms so that no one can answer, and maybe you will learn something. Geometry guy 23:39, 4 October 2008 (UTC)
 * In other words, you are claiming that everyone in the physical sciences and physical applied mathematics is using a term (contravariance) that (in your tautological interpretation) is essentially meaningless. And you are accusing me of arrogance?  Please give a reference.  —Steven G. Johnson (talk) 00:54, 5 October 2008 (UTC)

Some thoughts about contravariance
I seem to fail understanding how contravariance is what formalizes "direction in space". Let E be the set of real-valued functions of a real argument which satisfy $$f'''(t) + f'(t) = t$$. Let V be the set of real-valued functions of a real argument which satisfy $$f'''(t) + f'(t) = 0$$.

V is a three-dimensional vector space over the real field; E is an affine space over V, as the difference of two elements of E is in V, and the sum of an element of E and one of V is in E. Let's define the inner product in V by $$\langle f, g \rangle = \frac{1}{2\pi} \int_0^{2\pi} f(t)g(t)\,\mathrm{d}t$$. (An orthonormal basis of V is then, for example, $$\lbrace \hat x, \hat y, \hat z \rbrace$$, defined by $$\textstyle \hat {x}(t) = \sqrt{2}\cos t,$$ $$\textstyle\hat{y}(t) = \sqrt{2} \sin t,$$ and $$\textstyle\hat{z}(t) = 1.$$) We can then define the "distance" between two elements of E as $\overline{PQ}$ = $\sqrt{&lang;Q &minus; P, Q &minus; P&rang;}$, and the "angle" $$\widehat{POQ} = \arccos \frac{\langle P-O, Q-O \rangle}{\overline{OP} \,\overline{OQ}}.$$

If P is an element of E, O is another element of E which I'll arbitrarily refer to as "the origin", and $$(\hat{x}, \hat{y}, \hat{z})$$ is an ordered orthonormal basis of V, then I can define the coordinates of P as $$(P_x, P_y, P_z) = \left ( \langle P - O, \hat{x} \rangle, \langle P - O, \hat{y} \rangle, \langle P - O, \hat{z} \rangle \right )$$. Suppose you do the same with another "origin" of E, O&prime; and another ordered orthonormal basis of V, $$(\hat{x}', \hat{y}', \hat{z}');$$ you will obtain, for the same P, another triple of coordinates, say $$(P_{x'}, P_{y'}, P_{z'}).$$

Now, let S a function from R (the real numbers) to E; its derivative will be a function S&prime; from R to V. Let's define v = S&prime;(0) &isin; V. I will define the components of v the numbers $$(v_x, v_y, v_z) = \left ( \langle v, \hat{x} \rangle, \langle v, \hat{y} \rangle, \langle v, \hat{z} \rangle \right )$$. I might obtain the same result as $$(v_x, v_y, v_z) = \left. \left ( \mathrm{d}S_x(t)/\mathrm{d}t, \mathrm{d}S_y(t)/\mathrm{d}t, \mathrm{d}S_z(t)/\mathrm{d}t \right) \right |_{t=0}$$, where Si(t) (i = x, y, z) is the i-"coordinate" of S(t) as defined above for P. You do the same using your origin of E and ordered orthonormal basis of V, and obtain $$(v_{x'}, v_{y'}, v_{z'})$$. We find that: If I understand correctly, this is the property you would express as "v is a contravariant vector", yet v is just a real-valued periodic function with no harmonics higher than the first, so it has no relation whatsoever with directions of space such as north, down, right, etc. So, where am I wrong? -- Army1987 (t — c) 23:21, 10 October 2008 (UTC)
 * if we use the same orthonormal basis, we will find the same components of v, regardless of which origin we use for E;
 * if $$\begin{pmatrix} \hat{x} \\ \hat{y} \\ \hat{z} \end{pmatrix} = A\begin{pmatrix} \hat{x}' \\ \hat{y}' \\ \hat{z}' \end{pmatrix}$$ for some matrix A, then $$\begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix} = A\begin{pmatrix} v_{x'} \\ v_{y'} \\ v_{z'} \end{pmatrix}$$.
 * For a contravariant vector, I think you'll find that you need
 * $$(v_x \; v_y \; v_z) = (v_{x'} \; v_{y'} \; v_{z'}) A^{-1},$$
 * so that
 * $$(v_x \; v_y \; v_z) \begin{pmatrix} \hat{x} \\ \hat{y} \\ \hat{z} \end{pmatrix} = (v_{x'} \; v_{y'} \; v_{z'}) A^{-1} A \begin{pmatrix} \hat{x}' \\ \hat{y}' \\ \hat{z}' \end{pmatrix} = (v_{x'} \; v_{y'} \; v_{z'}) \begin{pmatrix} \hat{x}' \\ \hat{y}' \\ \hat{z}' \end{pmatrix} = v$$
 * Jheald (talk) 11:23, 12 October 2008 (UTC)
 * Both bases were assumed to be orthonormal, so A is an orthogonal matrix, and
 * $$\left( \begin{pmatrix}v_{x'} & v_{y'} & v_{z'}\end{pmatrix}A^{-1} \right)^{\mathrm T} = (A^{-1})^{\mathrm T}\begin{pmatrix}v_{x'} & v_{y'} & v_{z'}\end{pmatrix}^{\mathrm{T}} =A\begin{pmatrix} v_{x'} \\ v_{y'} \\ v_{z'} \end{pmatrix}.$$
 * In the most general case, you're right, A should be (A−1)T (and the way to retrieve "coordinates" of P would be more complicated), but I still fail to understand how that formalize the notion of a direction in physical space, as it applies to a somewhat abstract space as well. -- Army1987 (t — c) 12:23, 12 October 2008 (UTC)
 * As I've recently posted elsewhere below, I don't think this has any special relation to an idea of "simple physical spaces" either -- nor is it any reason for the article to say anything which doesn't generalise to larger and more general spaces.
 * As to A and being orthonormal, it only really makes sense to make a big song and dance about covariant and contravariant vectors in connection with basis transformations that are not locally orthonormal. Only then is it that the difference between covariant and contravariant really become apparent.
 * But in fact, I don't see that covariant/contravariant has anything much to do with the basic idea of vectors per se (arrows in a space) at all. It only comes in when you introduce reference bases; and only really becomes relevant, in terms of there being a distinction between covariant and contravariant when those reference bases are non-orthonormal.  That shouldn't come into the article until the very last sections, if at all. Jheald (talk) 15:01, 12 October 2008 (UTC)
 * Minor correction. I think it's probably best to write the contravariant and covariant transformations in their more typical forms,
 * $$\begin{pmatrix} v_{x'} \\ v_{y'} \\ v_{z'} \end{pmatrix} = A^{-1} \begin{pmatrix} v_{x} \\ v_{y} \\ v_{z} \end{pmatrix}$$
 * $$\begin{pmatrix} \hat{x}' & \hat{y}' & \hat{z}' \end{pmatrix} = \begin{pmatrix} \hat{x} & \hat{y} & \hat{z} \end{pmatrix} A,$$
 * so that vectors are represented by columns in each case. This underlines, even for orthonormal vectors, the difference between vectors which transform contravariantly, like v, and vectors which transform covariantly, like x-hat.  Whilst formally this is equivalent to what you've written, I find it easier to think of x-hat as a column vector than a row vector.  Jheald (talk) 21:38, 12 October 2008 (UTC)

Right-handedness
On a somewhat related note, how does one define right-handedness without referring either to the right hand or to a clock? All definitions of the right hand rule I've seen just refer to the fingers of the right hand. This seems hand-wavy in much the same way as saying "a thing with magnitude and direction". MarcusMaximus (talk) 04:12, 3 October 2008 (UTC)
 * Please, let's not have this discussion here. Start another section on the arbitrariness of sign conventions if you must, but can we first agree what this article is about?  —Steven G. Johnson (talk) 04:38, 3 October 2008 (UTC)
 * Have a look at orientation (mathematics). If you have questions, ask there&mdash;I agree that this is not the right place for that discussion. Ozob (talk) 18:53, 3 October 2008 (UTC)

I was posing the question because it is an important property of vector bases and the definition of the cross product. I thought it might be worth it to mention something in this article, but apparently you guys disagree. MarcusMaximus (talk) 21:30, 3 October 2008 (UTC)

What is this article about? (redux)
We need to decide what this article is about once and for all. The entire talk page is filled with arguments about what the article is about, what it should contain, and so on. This is ridiculous; the article needs to be about one thing, and we need to figure out what that one thing is.

Looking over the table of contents, I find that the article currently has: Almost all of this is very elementary stuff! A lot of this is secondary school material, and almost all the rest would be covered in vector calculus courses. Furthermore, it's done in R3 with the standard orientation and inner product, except for those cases where it's also done in R2. The only material that's not very elementary is the stuff on derivatives, pseudovectors, and contravariance.
 * A vague definition
 * How to write a vector in coordinates
 * Basic vector operations:
 * Adding
 * Scalar multiplication
 * Dot products
 * Cross products
 * Triple products
 * Bases and changing bases
 * Derivatives with respect to a vector (wrongly labeled the "derivative of a vector")
 * Examples of vectors in physics
 * Derivatives as elements of the tangent space
 * Pseudovectors and contravariance

I think the article should be about R2 and R3 with the standard basis, orientation and inner product. Derivatives, pseudovectors, and contravariance should get at most a sentence each redirecting the reader to more advanced articles: Directional derivative, pseudovector, and covariance and contravariance of vectors. I'm not convinced the material on tangent spaces is appropriate at all.

Steven, I know your finger is already itching to click "edit" so that you can disagree with me. But consider the intended audience of this article: Students. They may not have had any physics classes, ever. There are plenty of college students have trouble with the basic facts I'm proposing to keep in this article. I'm sure you've taught vector calculus; surely you've had students confuse vectors and scalars. Or not understand that vectors aren't bound to the origin. Or believe that the inner product of two vectors was another vector. The article is about basics like these for students like them, and nothing more. Ozob (talk) 00:58, 5 October 2008 (UTC)


 * The problem I have is that what you are proposing as the definition of these vectors (any two or three dimensional vector space over the reals with a Euclidean norm) is nonstandard: it is not how the original elementary concept of "magnitude and direction" "arrows in space" vectors is made precise in authoritative references (as opposed to definitions that are explicitly generalizations of the elementary vector concept...and we already have other articles on these generalizations). I thought that the goal in Wikipedia was to stick to what can be attributed to authoritative references?  We shouldn't be arguing about our opinions and interpretations, we should be arguing over how best to synthesize what can be found in reputable sources.  —Steven G. Johnson (talk) 01:15, 5 October 2008 (UTC)


 * The goal in Wikipedia is to (1) stick to what is verifiable and (2) to keep articles accessible with preference for readers and laypersons over editors and specialists.--Loodog (talk) 01:20, 5 October 2008 (UTC)


 * Everything I've said about the formal definition of these vectors is easily verifiable, and I already gave an authoritative reference (and many more could be found). Furthermore, I have never suggested that the entire article be written at this level, merely that one subsection give a brief and elementary (coordinate-based) definition ("having a direction" means that the vector components transform like the spatial coordinates under rotations).  What Ozob suggested is not verifiable as far as I can tell, because his suggested formal definition of the elementary "magnitude and direction" "arrows in space" vectors is nonstandard.   And we have to give some precise definition (as evidenced by the endless confusion and debate over the subject of this article).  —Steven G. Johnson (talk) 01:23, 5 October 2008 (UTC)


 * As to references Feynman in the Red Book I-11, introduces vectors being different to scalers in that they have direction, defines them as triples of numbers wrt a given basis, discusses how quantities they represent are invarience under translation and rotation of axis, defines the standard operations through coordinate representation, discusses derivatives wrt time, then introduces length. Its only very much later I-52 where he discusses polar and axial vectors but does use the terms contra and covarient vectors. To a large part we seem to be following Feynman quite closely. --Salix (talk): 08:01, 5 October 2008 (UTC)
 * But he presents contravariance as a more precise definition of the earlier concept, exactly like I said&mdash;both historically and pedagogically, "vectors" are first introduced in a fairly vague way as triplets of numbers representing a "direction" appealing to the intuition, and only later are made precise. Again, I'm not suggesting that we don't write most of the article at the more elementary level, merely that we have one subsection explaining that the notion of "having a direction" has a more precise standard meaning.  —Steven G. Johnson (talk) 14:27, 5 October 2008 (UTC)
 * That seems a reasonable suggestion. I think its also worth distinguishing between scalers (quantities without direction) and vector (quantities with direction) in such a section. --Salix (talk): 15:45, 5 October 2008 (UTC)

I am not proposing anything non-standard. In fact, I have to disagree with something that you seem to take for granted, Steven: The idea that a non-contravariant vector can possibly represent a direction. At the very elementary level that most of this article is written at, this is false. As far as this article is concerned, space is R3 with a fixed basis. Space has a distinguished point, the origin, and distinguished directions given by the standard basis. A direction in space is the same thing as a point, because all points are identified with vectors.

I grant immediately that this is not a productive viewpoint for more advanced work. But this article is not advanced, it's elementary. The article should be, as I said, about R2 and R3 with their standard basis, orientation, and inner product. There are lots of references for this; any multivariable calculus book will talk about vectors at this level (e.g., Stewart), and I'm sure that I could find high school textbooks that do, too. You seem to like citing Arfken and Weber, but that's a much, much more advanced book. Is there a reference that discusses contravariant vectors at a level appropriate for high school students? I doubt it, and that's why I believe that this material is inappropriate for this article. It should be at covariance and contravariance of vectors. (By the way, I've added a link to that article to the "See also" section; there was also a link much higher up, but since I believe the concept is important I think a separate link is appropriate). Ozob (talk) 01:22, 6 October 2008 (UTC)


 * First, if you look at high school textbooks etc, they generally define vectors as "magnitude and direction" "arrow in space". That is, they appeal to the intuition, and do not give a precise definition at all, and hence are not useful references for a precise definition.  (Note, by the way, that contravariance is implicitly important even for multivariable calculus, because otherwise things like the divergence of a vector field depend on the coordinate frame, and the usual "quiver plots" of vector fields make no sense unless they are contravariant.) In any case, even elementary treatments typically present the coordinate frame as an arbitrary choice (that the vector has some conceptual "graphical" existence independent of the coordinate frame), so requiring a fixed basis right off the bat is inconsistent with most elementary treatments.  We clearly need to give a precise definition somewhere, even if most of the article is at a handwaving level, and we clearly should make an attempt to give the standard precise definition.


 * Second, I agree that one could say that a vector space defines a "direction" in its "own" space, although this "definition" of "direction" is rather circular to say the least! However, that is irrelevant because this is is not how "having a direction" is defined in practice (in textbooks that do give a precise definition, excluding handwavy treatments and excluding abstract generalizations that don't define vectors in terms of "having direction" at all).  We shouldn't come up with our own definition here!


 * Third, all I am arguing for is one subsection that explains that there is a more precise definition, what motivates this definition, and gives the simplest (coordinate-based) version of this definition. I agree that most readers at a high-school level will be advised to stick to the hand-waving definitions.  However, the coordinate-based definition is certainly within reach of a freshman or even a smart high-school student.  See the explanation I gave above, which is basically two paragraphs, and is only a small fraction of the treatment in Arfken & Weber (which I cite because it is an authoritative reference, not because we should explain at their level). (See also Feynman, vol. I, sec 11-4, which gives a very similar coordinate-rotation based definition at an elementary level, if you don't like Arken & Weber.)  Pointing to the covariance and contravariance of vectors article is not an appropriate substitute here because it treats the subject for arbitrary curved manifolds (as it should, although it could be clearer!), which is much more complicated.  It's standard procedure in Wikipedia to have a short summary in an article like this that desperately needs a minimal precise definition phrased at the most basic possible level (editors have been arguing over what this article is about for years now), and a pointer to a main article elsewhere for a more complete treatment.


 * —Steven G. Johnson (talk) 20:18, 6 October 2008 (UTC)


 * I agree that high school textbooks and introductory calculus textbooks do not give precise definitions of vectors. I want this article to be more accurate than they are while still remaining on that level.


 * First, I disagree that contravariance is important for multivariable calculus. It's the chain rule which is important; the chain rule is what allows you to change coordinates. (More precisely, it's what allows you to compute derivatives in one set of coordinates with respect to another; in the abstract setting, this is of course a change of coordinates on the tangent bundle.) Change of frame is a special case of that which is insufficient for many applications: For example, you cannot convert from rectangular to spherical coordinates by making a change of frame. (I mean this both in the naive sense that the transition functions are not linear and also in the less naive sense that the induced maps on tangent or cotangent spaces are not special orthogonal.) I agree that the formula for divergence (as well as the gradient, the curl, the inner product, and so on) all depend on the choice of coordinates, but so what? One has to make a choice somewhere. Even if you try to introduce the dot product synthetically using the angle &theta; between the two vectors, the angle depends on a choice of inner product so the definition is circular. Eventually you have to fall back on a choice of basis. When a different coordinate system is needed&mdash;whether it's a change of frame or something more complicated&mdash;one makes substitutions (like r2 = x2 + y2 + z2 and so on) and applies the chain rule where needed.


 * You seem to consider SO(3) as special in some way that is not coming through to me. I don't see why you care so much about how vectors transform with respect to this group instead of with respect to an arbitrary coordinate change (that is, arbitrary diffeomorphism). If one identifies space with R3 (which is certainly done at this level&mdash;I agree that many textbooks will mention that the frame is arbitrary, but it's only mentioned in passing and is then ignored. Certainly it doesn't get through to the students) then changing coordinates from rectangular to cylindrical or spherical is important, and SO(3) loses its special place. I suppose that part of our disagreement is a difference of viewpoints. When I think of changes of coordinates for vectors, I'm thinking in terms of diffeomorphisms of open sets and their induced maps on vector bundles. (I'm not trying to be fancy; when I want to think precisely about what's going on, that's really what I have in my head.) From this point of view, the emphasis on SO(3) seems bizarre; it's only when one tries to relate to the real world that SO(3) begins to look natural, but even then I don't think I fully appreciate it. So I suppose I should ask straight: Why is change of frame so important to you? Why not SL(3) or GL(3)?


 * Second, if a vector is a "quantity with length and direction" then it must define a direction somehow. That direction would be the direction it points in; no contravariance is necessary here, only an identification of space with R3.


 * Third, I agree that the material at covariance and contravariance of vectors is currently much more advanced than the material here. But one could write a more elementary treatment at the start of the article and then get into technicalities later. Ozob (talk) 02:42, 7 October 2008 (UTC)


 * People talking about abstract vector spaces don't define it as a "quantity with length and direction". In the contexts where "having a direction" is considered part of the definition of having a vector, contravariance seems to be the standard way to formalize this.  As for the emphasis on SO(3), I'm not going to respond to your personalizing this&mdash;it's not what I think, this is the standard definition; go read a few books for the justification.   Anyway, I'm sick of arguing here, as the same arguments have circled about for (literally) years already here.  This article is doomed to endless controversy about what it is about, because it will always be dominated by editors who don't appreciate that "having a direction" has a standard formal definition (which most authors in the physical sciences consider rather crucial) separate from generalizations to abstract vector spaces.  —Steven G. Johnson (talk) 23:38, 11 October 2008 (UTC)


 * I took your advice and spent some time in the library today.


 * Arfken and Weber describe a contravariant vector in two ways. First, they say that it's a thing that transforms in a certain way with respect to SO(3). The particular type of thing is never explicitly described; it's given implicitly by how its components transform. Second, they call it a rank one tensor. They define tensors in a very classical way, as quantities whose components transform in a certain way, so as they say this is really the same as the first definition. In their definition of rank two tensors they don't seem to make any restriction on the possible coordinate changes. At the end of chapter two, there's a discussion of non-Cartesian coordinate systems which suggested to me that contravariant vectors transform as you might expect under non-orthogonal coordinate changes, too, though they never explicitly say so.


 * Next to Arfken and Weber was Boas, M., Mathematical methods in the physical sciences. Boas specifies that contravariant vectors transform contravariantly under arbitrary change of coordinates. So does Chow, T., Mathematical Methods for Physicists, p. 48 (he calls orthonormal coordinates "convenient in practice", p. 14), as well as Dunning-Davies, I., Mathematical Methods for Mathematicians, Physical Scientists, and Engineers, p. 391. Arfken and Weber seem to be the exception; all the other books I saw that defined contravariant vector made their vectors contravariant with respect to arbitrary coordinate changes, not just special orthogonal ones. That definition makes a contravariant vector an element of a tangent space. I am going to update the article accordingly. Ozob (talk) 22:53, 12 October 2008 (UTC)


 * I don't think there is any consensus for such a change. If not in this thread, then elsewhere, Steven has objected to characterizing a vector as an element of the tangent space.  I also object to this as unnecessary machinery.  If you want to think of vectors in terms of covariance and contravariance, then that is fine.  But that doesn't require you to think of them as elements of a tangent space.  For instance, there is no natural way to make the force an element of a tangent space (the units are different).  <font color="#c00000">siℓℓy rabbit  ( <font color="#c00000">talk ) 23:02, 12 October 2008 (UTC)


 * Furthermore, the subject of this article is vectors, not contravariant vectors. The basis vectors that transform covariantly are still vectors. Jheald (talk) 06:46, 13 October 2008 (UTC)


 * Indeed the discussion has gone on two long. Do you fancy writing a section on covariance and contravariance which would result in somthing productive from the discussion. --Salix (talk): 07:09, 12 October 2008 (UTC)
 * What is wrong in my above? -- Army1987 (t — c) 10:48, 12 October 2008 (UTC)


 * I have to say, I think this discussion is a nonsense. We already have a section Vectors, pseudovectors, and transformations, and that IMO is the right place for a discussion of covariance and contravariance. (Note, by the way, that the article Covariant transformation gives a much more accessible introduction to this at an introductory level than covariance and contravariance of vectors). Also, IMO the discussion certainly shouldn't come any higher up the article than that section is placed at the moment -- viz. the very last section of the whole article.


 * In relation to Stevenj's apparent assertion that "having a direction" is only meaningful, and only used in the literature, in the context of everyday physical spaces somehow distinct from any other Rn, or in discussions of covariance and contravariance, I put it to him that that simply is not true. It is certainly not true in numerical work -- for example, it is very common to talk of "singular directions" when taking the SVD of a matrix, or "eigendirections" when doing an eigendecomposition.  So, for example, numerical weather forecasting centres around the world routinely estimate the SVD of the 2-day ahead propagator of the current state of a global climate model to find the most singular directions in that decomposition.  A "direction" in that case typically corresponds to an entire weather system becoming more or less pronounced -- eg an entire anticyclone appearing over the Atlantic.  Similarly in pattern recognition, methods like principal component analysis identify "directions" in the feature space that capture as much as possible of the observed variation.  Or take the case you raised earlier, the charge distribution on a 2d surface, which one might capture with a finite elements approximation.  In a bifurcation analysis, one might look for the eigendirections corresponding to a particular eigenvalue moving through zero.  In this case the eigendirection would be an entire vector of local changes in the charge distribution, identifying the "direction" of a whole new solution splitting off.


 * You are I think saying that "direction" only makes sense to define something which persists under a co-ordinate transformation, and can be re-described in new transformed language. But whenever we have a vector that we can express according to a particular basis, we can always change basis and re-express the vector according to the new basis.  So in this sense a vector does always define a direction - whether the direction is identified in terms of spherical harmonics to express a particular charge density distribution over the surface of a sphere, or identified in terms of whatever other set of basis directions you happen to be interested in.  Jheald (talk) 14:01, 12 October 2008 (UTC)


 * Thinking some more, the problem is maybe in seeing "length + direction" as the definition of a vector. "Length + direction" is certainly quite useful at an intuitive, introductory level; and particularly useful to distinguish vectors from scalars.  At a more rigorous level, perhaps, "direction" is an important property that can always be associated with a vector (at least in finite dimensions); but the definition of a vector becomes being an element of a vector space, with "direction" best defined only once you have already established the vector space and its vectors.  Jheald (talk) 14:01, 12 October 2008 (UTC)

Bound and free vectors
I don't understand the significance in this article of distinguishing "bound" and "free" vectors by saying that bound vectors have their tail at the origin and (implying) that free vectors have their tail fixed at some other arbitrary point. The origin of a coordinate system is also an arbitrary point, so there seems to be no qualitative difference here.

What does it mean for a vector's tail to be "fixed to the origin" anyways? The only time that makes sense to me is if you are talking about a displacement vector from the origin to a point (but again, this is not qualitatively different from a displacement vector between any two points). The other case is the original member of a vector field where every point in space has a unique vector associated with it. However, velocity vectors, acceleration vectors, angular velocity vectors, angular acceleration vectors, and torque vectors have no need for a location in space, and it usually doesn't even make sense to assign them a location.

This article used to say that you weren't even allowed to add two free vectors unless they had the same base point. I removed that statement, because it disqualifies all kinds of important vector equations, most obviously the simple addition of the displacement vectors rAB and rBC to find the displacement rAC.

Kane uses different definitions for bound and free vectors. He says that free vectors are vectors without a specific location, while bound vectors have at least a line of action parallel to the vector and passing through a specific point (like a force on a rigid body, or a member of a vector field). Obviously a displacement vector between two points has both ends fixed.

Is there a good explanation for the definitions used in this article that I'm just not aware of? MarcusMaximus (talk) 07:30, 9 October 2008 (UTC)


 * There was some inconsistency with the way the article used the two terms. I have made a few changes in attempt to fix the problem.  <font color="#c00000">siℓℓy rabbit  ( <font color="#c00000">talk ) 12:38, 12 October 2008 (UTC)

I agree with your edits. I have a question though. The article used to state "If the Euclidean space is equipped with a choice of origin, then any free vector is equivalent to a bound vector whose initial point is the origin." I didn't think this was precisely true, so I changed it to "If the Euclidean space is equipped with a choice of origin, then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin."

But what does it mean to say that a free vector and a bound vector are equivalent? Can you give an example of a free vector (say, velocity) that could have an "equivalent" bound vector? Perhaps a generic free displacement vector could be considered equivalent to the bound displacement vector between two points? But in that case, the origin is not particularly interesting. MarcusMaximus (talk) 08:13, 13 October 2008 (UTC)


 * I had hoped to avoid an explicit discussion of equivalence classes, particularly this early in the article. A free vector is an equivalence class of bound vectors, where $$\overrightarrow{AB}$$ and $$\overrightarrow{A'B'}$$ are equivalent if and only if ABB&prime;A&prime; is a parallelogram.  Now, if a is a free vector and O is any marked point of the Euclidean space, then there is a unique bound vector in the equivalence class defining a whose initial point is at O. <font color="#c00000">siℓℓy rabbit  ( <font color="#c00000">talk ) 12:32, 13 October 2008 (UTC)

I think what you have just typed here is sufficient. It doesn't even need to mention equivalence classes. I also appreciate what you have done to de-emphasize the "origin of a coordinate system" and framed the discussion in terms of arbitrary points. A coordinate system is superfluous for a large portion of the vector properties described in this article. MarcusMaximus (talk) 02:59, 14 October 2008 (UTC)

Do we need an article about vectors in physics?
With the title "Euclidean vector" and all this talk of abstract things such as contravariance (which I still fail to understand how it can formalize the concept of "directions" such as "left", "right", "southwards" and so on), this article is essentially about a mathematical concept. Do we need an article called Vector (physics)? See User:Army1987/Vector_(physics) for what its lead would look like. (I apologize for the really lousy prose, but it's just an example, I wrote it in five minutes.) -- Army1987 (t — c) 16:18, 12 October 2008 (UTC)
 * I think the present article, as it is currently, gives a much better elementary description of vectors of the kind that are used in physics. What is it you feel is wrong or inappropriate about the present article, as it is at this moment, for an audience of physicists at quite an introductory level ?  Jheald (talk) 16:32, 12 October 2008 (UTC)

Pseudovectors
It seems to me that the way pseudovectors are currently discussed in the Vectors, pseudovectors, and transformations section of the article could use a revision.

Immediately after a discussion of vectors transforming contravariantly under coordinate transformations (passive transformations), IMO it's unfortunate that pseudovectors are then presented in terms of an active transformation, introduced by the statement that


 * Some vectors transform like contravariant vectors, except that when they are reflected through a mirror, they flip and gain a minus sign.

Can I suggest it would be much better if this section was re-written so that rather than presenting, as it does at the moment, an active transformation, (the car itself reflected in the mirror), the consequences are investigated of a parity inversion that is a passive transformation (ie reflecting the co-ordinate system in a mirror, but leaving the car unchanged)?

Initially we start with a right-handed system, with for example x pointing straight ahead, y to the left, and z vertically upwards. Reflecting in the xz plane gives a left-handed co-ordinate system, so x still points ahead, but the y axis now points to the right.

The co-ordinates of the centre of the left hand wheel, which were (0,1,0) now become (0,-1,0). Contravariant vectors transform like co-ordinates, so this is also how contravariant vectors transform.

The angular momentum of the wheel is given by a bivector in the plane $$\hat{z} \wedge \hat{x}$$. In the right-handed co-ordinate system, conventionally this bivector is represented by a vector in the direction $$\hat{z} \times \hat{x}$$, ie a vector parallel to (0,1,0), so pointing to the left. In the left-handed co-ordinate system, the bivector is still represented by a a vector in the direction $$\hat{z} \times \hat{x}$$, ie a vector parallel to (0,1,0); but in the new co-ordinate system this now designates a vector pointing right.

A true vector is therefore an arrow which remains invariant under an reflection of the co-ordinates (a passive transformation). But a pseudovector is an arrow which flips direction according to the handedness of the co-ordinate system.

Angular velocity, magnetic field, and torque are examples of such pseudovectors.

... I think such a discussion, considering the reflection as another passive transformation, more accurately presents what physicists understand by the term; and follows on naturally from the previous discussion of contravariant vectors.

In this way the components of a pseudovector thus transform like a contravariant vector, but gain an additional factor of -1 under co-ordinate reflections. Jheald (talk) 12:32, 13 October 2008 (UTC)


 * This is a problem endemic to much of the material on contravariance and covariance on Wikipedia. One can only meaningfully speak about the co(tra)variant components of a vector relative to a reference frame, or at least to a system of reference frames under consideration.  Arfken and Webber, as well as most treatments of vector analysis, are careful at least initially to do this.  <font color="#c00000">siℓℓy rabbit  ( <font color="#c00000">talk ) 12:48, 13 October 2008 (UTC)


 * I have to say, the continued stress on the word components doesn't quite chime for me. A contravariant vector is a whole arrow, that's moving to another whole arrow, not individual components.  The important point is that it's not the arrow in physical space that's moving, it's an arrow moving in an "auxiliary linear space V", as you've just defined it elsewhere -- the representation of the physical arrow, if the mathematical language police allow us to use that word (I'm a bit nervous, having been told off at spinor).


 * So I'd much rather talk about the componentwise representation of the vector transforming contravariantly, than the "components" of a vector doing so.


 * That said, that is not really the issue I was getting at here. This discussion in the last two paragraphs is all about passive transformations - transformations of the co-ordinate system, leaving the physical reality invariant.


 * But active transformations, mapping physical reality to a physically different state, are also something that one can discuss - something different, but still entirely legitimate. As it is at the moment, the article quite accurately discusses what happens to a pseudovector under an active transformation.


 * My point is that it would be more appropriate for the article to discuss what happens to pseudovectors under passive transformations (ie changes purely in the co-ordinate system) - both in terms of article flow, since the article has just been discussing passive transformations to discuss contravariant vectors; and because this reveals that a purely notational change can cause the "arrow" corresponding to a pseudovector quantity to reverse direction in the physical space - conveying the idea that arrows in space may not be the best way to think about what are really bivector quantities. Jheald (talk) 15:01, 13 October 2008 (UTC)

should something be said about vector division?
? 212.200.243.13 (talk) 00:33, 26 March 2009 (UTC)


 * What do you want to say about vector division? —Ben FrantzDale (talk) 13:06, 26 March 2009 (UTC)


 * It doesn't work. The end. Ozob (talk) 14:48, 26 March 2009 (UTC)


 * it should be said! 212.200.243.13 (talk) 00:21, 27 March 2009 (UTC)


 * Not so fast. There is the geometric product of vectors, which can be inverted. —Ben FrantzDale (talk) 02:52, 27 March 2009 (UTC)


 * That takes place in a larger object, a Clifford algebra. I stand by my previous statement: There is no way to make R3 a real division algebra. See Frobenius theorem (real division algebras). Ozob (talk) 16:41, 27 March 2009 (UTC)


 * There should definitely be a note about division in the article. People are used to that addition is complemented by subtraction, and likewise, multiplication by division. Absence of division related paragraph may imply that it is trivial, and just an inverse operation. 212.200.243.13 (talk) 08:35, 28 March 2009 (UTC)


 * from Wolfram 212.200.243.13 (talk) 18:42, 28 March 2009 (UTC)

Can someone provide a rational argument why division should not be mentioned here when it can be referenced to other existing printed mathematical encyclopedias? 212.200.243.13 (talk) 09:05, 5 April 2009 (UTC)
 * In general, division isn't well defined. I have no problem mentioning this fact and referring to generalizations which allow for vector division such as Clifford algebra. Be bold. —Ben FrantzDale (talk) 19:16, 5 April 2009 (UTC)


 * although i am bold, i'm in no mood for edit war, and a.r. reverted my addition without any argument but his personal opinion. 212.200.243.13 (talk) 20:36, 5 April 2009 (UTC)

Vector addition and subtraction: Tautological definition.
The article has this to say about adding vectors together, which isn't terribly useful to a reader who doesn't already know how to add vectors together:

Assume now that a and b are not necessarily equal vectors, but that they may have different magnitudes and directions. The sum of a and b is
 * $$\mathbf{a}+\mathbf{b}

=(a_1+b_1)\mathbf{e_1} +(a_2+b_2)\mathbf{e_2} +(a_3+b_3)\mathbf{e_3}.$$

Evaluating that equation yields more vectors to add together:

\mathbf{a}+\mathbf{b} =(a_1+b_1,0,0) +(0,a_2+b_2,0) +(0,0,a_3+b_3) $$

How do you add these vectors together? Well obviously, by applying the rule given by the article:



(a_1+b_1,0,0) +(0,a_2+b_2,0) +(0,0,a_3+b_3) =(a_1+b_1+0+0)\mathbf{e_1} +(0+a_2+b_2+0)\mathbf{e_2} +(0+0+a_3+b+3)\mathbf{e_3} $$

Which reduces to:
 * $$\mathbf{a}+\mathbf{b}

=(a_1+b_1)\mathbf{e_1} +(a_2+b_2)\mathbf{e_2} +(a_3+b_3)\mathbf{e_3}.$$

In other words, to add some vectors together, you must first add some vectors together.

One possible non-tautological description of vector addition that seems to make sense (but I am no mathematician) would be this:

$$\mathbf{a}+\mathbf{b}=(a_1+b_1, a_2+b_2, a_3+b_3)$$

71.72.235.91 (talk) 00:35, 23 October 2009 (UTC)


 * I disagree. While it might be semantically a tautological definition, in reality it's not.  Because the basis vectors have been introduced as an alternate means of vector representation, they are not here to be vectors and no one but a computer would take the distributive step you did initially, especially because no where else in the article is this kind of distribution explicitly mentioned.  Also, if you look through the article, this is consistently the notion used. It's just a notation. And a common one at that. I mean:


 * $$(2,5,3)=2\mathbf{e_1}+5\mathbf{e_2}+3\mathbf{e_3}$$


 * By your reasoning, the right-hand side would be completely useless since now we have to go plug in what e1,e2, and e3 are, which no one ever does.--Louiedog (talk) 00:09, 25 October 2009 (UTC)


 * Maybe that right-hand side makes it easier for Ph.Ds to prove theorems, but it makes it much more difficult to figure out how to do the basic vector operations if this is your first attempt to learn how to work with vectors. For that purpose, the right-hand side is in fact completely useless.


 * And no, it's not "just a notation." An expression like $$2\mathbf{e_1}+5\mathbf{e_2}+3\mathbf{e_3}$$ is not an atomic expression that can be trivially translated back into the form $$(2,5,3)$$. Just like you have to know how to add before you can call $$2 + 5$$ "just an alternate notation for $$7$$," you have to know how to add and multiply vectors before you can understand what $$2\mathbf{e_1}+5\mathbf{e_2}+3\mathbf{e_3}$$ means, or why $$(2,5,3)$$ is the same value. -- 71.72.235.91 (talk) 17:48, 26 October 2009 (UTC)
 * The same information is parseable from either side with the same amount of work. Unlike 5+2=7, there is no information changing operation necessary to convert from one notation to the other.  Every course I've ever had or taught that had vectors in it used (1,2,3) and i+2j + 3k notation interchangeably.


 * Besides, I could just as easily argue that
 * $$\mathbf{a}+\mathbf{b}=(a_1+b_1, a_2+b_2, a_3+b_3)$$
 * is tautological because the numbers in parenthesis are understood to be coefficients of basis vectors.--Louiedog (talk) 18:12, 26 October 2009 (UTC)

Radial Vectors?
Radial vectors are used in introductory physics courses, and even used in the wikipedia article on Newton's Universal Gravitation. But there is no information on wikipedia (this article inclued), or on the internet (with a quick google search on "radial vectors"), or even in my introductory physics book that says what radial vectors are. Our instructor gave us a quick pictorial explanation when he first used them, which seemed easy enough, but its necessarily geometric in nature, and doesn't show anything about how they relate to other objects, and what makes a radial vector different from a regular vector, nor does it really give a good definition of what they are. It'd be great if someone could remedy this, by perhaps making a radial vector article, or making a section in this article. 76.175.72.51 (talk) 22:53, 31 October 2009 (UTC)
 * The reason you're not finding "radial vector" here or on google is because you're using rather strange terminology that suggests you've organized the concepts in your mind in way that's different from how it is.


 * But I think I can tell what you mean by it. Vectors can point in any direction they please in the most general sense, which is the sense treated here.  Now, it might happen that in certain situations the vectors you end up using have radial symmetry, in which case, you're probably going to use spherical coordinates in your calculations because they're tailored for this kind of thing.  We don't have a special class of vectors called "radial vectors" that are somehow different from the rest.--Louiedog (talk) 20:53, 3 November 2009 (UTC)

So what is a *Euclidean* vector?
The article is called "Euclidean vector" but only offers definitions of a "vector" (and only gives examples of Euclidean vectors). Are we to regard "Euclidean vector" as synonymous with "vector"? If not, it would be helpful to define the "Euclidean" part of "Euclidean vector" and give some examples of vectors that are not Euclidean (rather than just a "see also" list). Dependent Variable (talk) 14:17, 14 December 2009 (UTC)


 * If you read the dusty tomes of the old discussions here, you find that the name has been a subject of tremendous debate. There is consensus that the present name is bad, but better than all the alternatives. Myself, I'd call one of these things "an element of the tangent bundle of R3", except that binds the vector to a base point, which is not quite what we're trying to get at. The important distinction between these and other vectors is that these have geometric content&mdash;they describe the real world. Abstract vector spaces don't a priori describe anything physical at all. Ozob (talk) 15:06, 14 December 2009 (UTC)


 * Do you think it's worth opening the article with something like "...a Euclidean vector (sometimes called a geometric or spatial vector, or simply a vector) is..." and making the link from "When it becomes necessary to distinguish it from vectors as defined elsewhere..." go directly to Vector space, rather than the disambiguation page Vector? (Assuming that the main other kind of definition of relevance to maths and physics is the abstract "an element of a vector space".) Dependent Variable (talk) 04:33, 19 December 2009 (UTC)


 * Yes, I think that's a good idea. In particular, vector space is definitely the right place for that link to go to. Ozob (talk) 04:40, 19 December 2009 (UTC)