Talk:Euclidean vector/Archive 3

The Gibbs-Heaviside Vector is a Quantity
With all due respect, a 1st year University level Physics vector (i.e., a Gibbs-Heaviside Vector) is absolutely a quantity.


 * Vectors are neither numbers nor quanities. They are quantitative but they are not scalars. Numbers are ordinals, cardinals, or measures (those links are to theoretical articles that do indeed make the topic seem more difficult than it needs to be). Quantitative objects, such as vectors, use numbers in their description; so for example a three-dimensional vector is typically represented as an ordered list of three numbers. But they are not numbers themselves; a pride of lions is not a lion (a Pride does not have four paws), a string of letters ("hello") is not a letter ("h"), and so forth. A vector is a compound object built with numbers. Pete St.John (talk) 20:41, 22 February 2008 (UTC)

Definitions of Vectors from a sample of Physics Texts
1) From Physics (5th Ed) by Douglas C. Giancoli c1998, Prentice Hall; ch.3, p.48

"A quantity such as a velocity, which has direction as well as magnitude, is a vector quantity. Other quantities that are also vectors are displacement, force, and momentum."

2) From The Feynman Lectures on Physics by Richard Feynman, Volume 1 c1963, Addison-Wesley; ch. 11, p.11-5

"[L]ike the number of potatoes in a sack, we call an ordinary quantity, or an undirected quantity, a scalar. Temperature is an example of such a quantity.  Other quantities that are important in physics do have direction, for instance velocity: we have to keep track of which way a body is going, not just its speed."

3) From The History of Physics by Isaac Asimov c1984, Walker and Company, NY; ch. 3, p 27

"A quantity that has both size and direction, as force does, is a vector quantity, or simply a vector. One that has size only is a scalar quantity."

4) From Conceptual Physics (8th Edition) by Paul G. Hewitt c1998, Addison-Wesley; ch. 3, p.37

"Any quantity that requires both magnitude (the amount of, or how much) and direction (which way) for a complete description is a vector quantity. Thus velocity is a vector quantity.  Other examples are force and acceleration.  By contrast, a quantity that can be described by magnitude only, not involving direction, is called a scalar quantity. Speed is a scalar quantity, as are mass and volume.  So speed is a scalar quantity and velocity is a vector quantity."

--Firefly322 (talk) 23:38, 22 February 2008 (UTC)

As for Vectors as Numbers, Some Other Wikipedia Entries related to Vectors
Complex number (which is what Gibbs-Heaviside vectors decended from via Hamilton's Quaternions).

Quaternion

hypercomplex number (the set of numbers to which Quaternions belong):

Holor (Gibbs-Heaviside vectors are classified as a type of univalent holor, scalars are classified as a type of nilvalent holor)

--Firefly322 (talk) 02:03, 23 February 2008 (UTC)


 * The holor article doesn't call holors either "numbers" or "quantities". Here's the quote: "A holor is a mathematical entity that is made up of one or more independent quantities". Also, the quaternion article doesn't describe them as numbers, but rather as an "extension" of complex numbers. Some other examples: tensor is an "object", and a mathematical vector is an "object". In Griffiths's E&M textbook, it's a "quantity". Like I said before, I believe that "quantity" is one of many legitimate and accurate ways to describe a spatial vector. The only question is whether it's the least confusing way. I suspect that the way most people are used to understanding the term is such that "geometric object" is likely to be less confusing to most (but not all) readers, when it's put in the first sentence like it is here. Speaking of which, would "mathematical entity" be a good alternative? --Steve (talk) 03:50, 23 February 2008 (UTC)

More on Vectors as Numbers
number

Here in this A-grade entry, Quaternions are indeed described as numbers:

''Sets of numbers that are not subsets of the complex numbers include the quaternions H, invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative. ''

Here's the logic:


 * Quaternions are numbers
 * A Gibbs-Heaviside Vector is type of Quaternion
 * Therefore, a Gibbs-Heaviside Vector is also a number

--Firefly322 (talk) 14:56, 23 February 2008 (UTC)

Propose that a Spatial Vector is completely defined by Gibbs's 3 vector system
To Gibbs in his vector analysis, space is completely defined by 3 orthogonal unit vectors (i,j,k). So in the lead-in the use of Spatial Vectors in light of comments from the well respected Moon (of MIT) and Spencer who stated about vector space "The 'space' is non-metric; and the 'vector' ... is completely different from the heavily restricted 'vector' of vector analysis [i.e., the school of analysis begun by Gibbs and independently and simultaneously created by Heaviside]." (Here's a link to Gibb's original writing.)

http://books.google.com/books?id=NV5KAAAAMAAJ&printsec=frontcover&dq=Elements+of+Vector+Analysis

Gibbs's vector anlaysis is consistent with Galilean-Newtonian views of the physical world around us. Our cars, our planes, our buildings, our bridges, and everything else that is macroscopic in size and not moving near the speed of light. Still, Maxwell's equations were originally visualized and manipulated in a 3 dimensional space. And they remain so today. Thus, both Newton's and Maxwell's laws of the universe are consistent with a 3 dimensional view of space. Hence the spatial vector should be defined as 3 dimensional (naturally, noting a 1 and 2 dimensional vectors as subsets) and finite-dimensions beyond 3 as being quite exceptional in theory and practice. Even the 4 dimensions of Einstein's spacetime are actually 3 spacial dimensions plus 1 temporal dimension, which are conceptually consistent with the 3 dimensional spacial view of Galileo, Newton, Maxwell, and Gibbs. --Firefly322 (talk) 17:52, 23 February 2008 (UTC)

--Firefly322 (talk) 15:44, 23 February 2008 (UTC)

A quote from Vectors by Moon and Spencer
Moon and Spencer analyze around 20 definitions for vector--from Gibbs to M.R. Spiegel to Wilson. They go on to state (p. 321)

"Evidently the 'vector' of tensor analysis has little in common with the 'vector' of vector analysis. The latter is definitely limited to Euclidean 3-space and may even require rectangular coordinates. We have seen that there are at least five definitions of 'vector' advocated in vector analysis. The word is defined as a quantity  (a) Having magnitude and direction,   (b) Specified by three numbers,   (c) Given by a directed line segment,  (d) Allowing parallelogram addition,   (e) Having certain transformation properties. Since these various defintions all act as an introduction to the same subject of vector analysis, one expects them to refer to the same entity. But this can hardly be the case. A quantity (b) specified by three numbers is merely a univalent holor [which is more or less a 1xN or Nx1 matrix]. Every time the coordinate system is changed, the three numbers change, and there is nothing in the definition to tell how they must change to make vector analysis a meaningful discipline.

The geometric definitions (a) and (c) perhaps imply vaguely the idea of a geometric object--the arrow--which is more basic than the coordinate system and which is invariant with respect to coordinate transformation. Such an idea may be implied but it certainly is not stated. And no indication is given as to the GROUP of transformations under which this arrow is to remain invariant. In fact, one finds the whole subject of vector analysis built implicity on one coordinate system. Gradient, divergence, and curl are usually defined only in rectangular Caresian coordinates, as if no other coordinates were ever necessary. Yet practical applications to FIELDS nearly always require curvilinear coordinates. So the ordinary treatment, which has based its proofs on rectangular coordiantes, is forced into highly questionable evasions when it attempts to patch up its systems for curvilinear coordiantes. An example occurs with the [DEL]-operator, as shown in appendix C.

Apparently sensing a lack of rigor, some mathematicians have introduced (d), which requires that the directed line segments shall add by parallelogram addition. Obviously, this modification does not touch the fundamental defect involving transformations.

Recently the word 'vector' has also become prominent in the term 'vector space'. This term has only the remotest connection with familiar meansings of either 'vector' or 'space'.

'A vector space is any set of geometric objects X, Y, Z, ... called VECTORS that can be 'added' to each other and 'multiplied' by real numbers a, b, c ...-the resulting sums and products being again vectors in the vector space--provided these abstract operations of addition and multiplication obey certain of the laws of ordinary arithmetic'. -- R.H.Crowell and R.E.Williamson (p.53 Calculus of vector functions, 1962, Prentice Hall)

The 'space' is non-metric and the 'vector', being closely related to the general univalent holor [which is more or less a 1xN or Nx1 matrix], is completely different from the heavily restricted 'vector' of vector analysis.

As a final example of the word 'vector', we might mention the electrical engineer's 'vector diagram'. It involves voltage and current or power, represented by directed line segments in a 2-space. It has no bearing on vector analysis; and fortunately the word 'vector' is now being replaced in this connection by 'phasor'." --Firefly322 (talk) 17:02, 23 February 2008 (UTC)


 * Would anyone object to putting the following in for the first sentence? "A spatial vector, or simply vector, is a tool used to describe quantities, such as displacement or velocity, which have both a magnitude and a direction". It's a bit physics-specific, but I think gives people the right idea, right off the bat. And I don't think the parallelogram law needs to be in the first sentence. --Steve (talk) 17:17, 23 February 2008 (UTC)


 * What is proposed sounds unobjectionable. But another, perhaps compatible idea would be for contributors to ultimately agree on a vision of this page by means of research and thought and persuasion.  I believe Silly Rabbit had a good start when he said the rule of thumb for a target audience should be that of first year physics students.  I think just removing the parallelogram from the first sentence would be enough and also make the sentence consistent with 1st year university physics texts.  spatial vector, or simply vector, is a quantity characterized by a magnitude and a direction is good, already matching the physics texts.  Later on in the article, after the lead-in quantities should be elaborated on. --Firefly322 (talk) 17:59, 23 February 2008 (UTC)

Now that (hopefully), things have cooled off, I have made some changes to the lead consistent with your suggestions. Please let me know if you approve. Silly rabbit (talk) 21:53, 23 February 2008 (UTC)


 * I would suggest the removal of row vector and column vector from the lead-in. These aren't quantities of Newtonian and Maxwellian physical laws.  These two things really are mathematical entities that can and do hold information not conforming to physical laws. --Firefly322 (talk) 22:34, 23 February 2008 (UTC)


 * The reason that I included them originally was to point to the articles row vector and column vector which treat explicitly the idea of a vector as a list-of-numbers. My rationale for including it was so that confused readers can more easily navigate through a bunch of related ideas.  Thus, if a reader is not interested in a spatial vector per se, then he or she can navigate directly to the relevant article.  However, I think a better way forward might be to create a sidebar where various related ideas are linked (a navbox).  I will set this up shortly. Silly rabbit (talk) 22:42, 23 February 2008 (UTC)


 * It sounds like a good idea. I'm all for it. I say do the sidebar. --Firefly322 (talk) 12:26, 24 February 2008 (UTC)

Regarding the beginning point (i.e., the vector's tail )
Now that the issue with the mathematical entities of row and column vectors has been improved. I see something else.

In both Newtonian and Maxwellian [Maxwell's equations have realitivity built-in, so Maxwellian is an efficient way I have including Einstein's realitivity] physics, besides a displacment vector (which has spatial coordinates for both tip and tail ) usually only the tail has spatial coordinates. For example, vectors of velocity, force, and acceleration have only 3 dimensional spatial coordinates for their tails. None of their tips are defined in 3 dimensional physical space.

Physically, this corresponds to an observer locating the endpoints of a vector with respect to a fixed system of measuring rods.

So it's physically impossible for an observer to spatially localize more than one endpoint for most types of physical vectors. And this sentence should be recast somehow. --Firefly322 (talk) 00:03, 25 February 2008 (UTC)


 * In light of recent editorial efforts, the following sentence now reads accurately from many points of view:


 * Physically, this corresponds to an observer making a finite number of measurements at a single spatial point with respect to a fixed system of measuring rods.


 * Yet the previous sentence (see below) with the mathematical viewpoint seems somehow weak in relation to this sentence with the physical viewpoint.


 * Let me elaborate on this Maxwellian-Heaviside exclusion principle, if you will, for physically measuring the tails and tips of vectors, so that someone might be able to translate it into a short, precise, and still easy-to-understand mathematical statement. Ideally, each measuring rod needs to have the same units as the vector it measures.  So a force vector must be measured by a force measuring rod, a current density vector by a current density measuring rod, and a speed-of-light vector must be measured by a speed-of-light measuring rod, for example.


 * The way I'm looking at it, the consequence is that every vector exists in two separate Finite dimensional spaces. One the physical space (where the finite number of orthogonal unit vectors are all distances of measurable length such as angstroms, miles, light-years, etc.) with each vector's tail at only a single spatial point (even displacement vectors, in this scheme of thought, would have their tips in the second space); the other a unit analysis measurement space where the tip of the vector exists as a directed, continuous mathematical interval (which conceptually feels like a single quantity and number, i.e. a vector quantity).  In this unit analysis space, the unit vectors are of the type of vector being measured. That is, for these examples, a force space (where 1, 2, 3, or more orthogonal unit vectors allow comparison with force by ideally being vectors of force themselves), a current density space (where 1,2, 3 or more orthogonal unit vectors allow comparison with current density by ideally being vectors of current density themselves ), and a speed-of-light space (where 1, 2, 3 or more orthogonal units vectors allow comparison with the speed-of-light by ideally being vectors of the speed-of-light themselves).


 * This conceptual soup has already been boiled down for the sentence with the first word "Physically" above. Now I believe there should be a way to boil it down again to a few nice phrases that can be placed into the following sentence:


 * Mathematically, this represents the vector as a list of real numbers giving its Cartesian coordinates. --Firefly322 (talk) 15:18, 25 February 2008 (UTC)

Principle of Least Action
A suggestion is to employ Leonhard Euler's principle of least action in the lead-in. Perhaps is a more universal principle than spacetime (i.e., its application across a wide variety of physical situations is much simpler than the application of any frame of space and time, Einstein's or Newton's.) --Firefly322 (talk) 00:01, 26 February 2008 (UTC)

Keep it focused
I don't think it's helpful to dump all sorts of things that are connected to vectors in one way or another into this (mathematical) article. I rewrote the lead trying to emphasize the basics of vectors. While vectors do have an enormous range of applications, the proper place for detailed discussion of them is almost certainly in the corresponding specialized articles. Furthermore, the function of the lead is to briefly summarize the content of the article. Thus for a highly developed and carefully construed article such as this one, any new topics should go into the body of the article first, and only if they integrate well, they should be given prominent attention in the lead. Ah, and to prevent unnecessary acrimony, here is a request to all new editors:


 * Please, be mindful of the tremendous work that has already gone into this article and respect the consensus reached. Try to understand the reason for why things the way they are before launching into discussions. And show consideration for editors like Silly Rabbit who have invested a lot of effort into making this a coherent and readable article.

Arcfrk (talk) 00:43, 26 February 2008 (UTC)


 * Just for the record: I object to turning this article into a manual on vectors for first-year physics students, as has been proposed above, or anything of that sort. This is a general purpose encyclopaedic article, and it should stay that way. Peppering the text with links to online notes from physics classes as "inline citations" is also inappropriate: vectors have been fairly uncontroversial in the last 70 years or so, so clear mainstream description does not require such references. Arcfrk (talk) 00:56, 26 February 2008 (UTC)


 * For mathematics, "vector" "space" is the place to go.  This article is, to paraphrase Mr. Silly a rabbit, a rare collaboration between physicists and mathematicians.  Although there seems to be few from the physical school of persuasion contributing.  Besides, as far as I know the lead was almost entirely written by Mr. Silly Rabbit.  --Firefly322 (talk) 01:29, 26 February 2008 (UTC)


 * You have strong opinions. Have you tried "what links here"? It's a very useful feature of Wikipedia, and gives a much better idea of how an article is actually used, not according to your individual "persuasions", but in an encyclopaedia that has been gradually by thousands of editors. You will see many, I mean MANY mathematics articles linking here. By the way, reverting constructive edits is a bad form, and WP:Sockpuppeting is a serious and punishable offense. Arcfrk (talk) 02:01, 26 February 2008 (UTC)


 * Just for the record, I prefer Arcfrk's version. Silly rabbit (talk) 02:05, 26 February 2008 (UTC)


 * Reverting your edit was a reasonable thing to do, since you made a major change and didn't discuss it first. If this is to be an article about mathematical vectors in a mathematical vector "space" that's fine, but that should made clear. --Firefly322 (talk) 07:33, 26 February 2008 (UTC)
 * Also the idea of a mathematical vector in vector space has been controversial since its widespread introduction 40 or so years ago. The citiation in Vectors by Moon and Spencer makes that clear.  No physicists would think that a vector as a quantity would be controversial, but most of the contributors to this article found it so.  I think you fail to understand the differences between physical and mathematical schools of thought.  And simply label what isn't in the mathematical camp, in spite of quotes and references from well-respected physics texts, as someone's opinion.
 * Finally, what arcfrk appears to have in mind is not a general article about vectors as found in 1st year calculus and 1st year physics texts. His vision is of an article that confuses the abstract, non-physical 1xN or Nx1 matrixes of linear algebra with the physical Gibbs-Heaviside vectors of introductory calculus and physics. --Firefly322 (talk) 07:52, 26 February 2008 (UTC)
 * I don't know of any controversy regarding vectors within the last 40 years; in 1968, Quantuum Mechanics had been employing Hilbert Spaces for about sixty years. The problem with the word "quantity" is semantics, and just the difference between jargons of different disciplines: mathematicians don't want to call vectors quantities because we want to distinguish them from scalars; that's why I prefer to say "quantitative object" instead of "quantity" to describe a vector. For other people it's natural to use "quanity" for "quantitative object", in physics that might be because the scalar field is generally not very interesting in itself, while to many mathematicians the scalar field is the object of study itself. So mathematicians are touchy about the words "number" and "quantity". But we can deal with this by being explicit: a three-dimensional vector is an ordered list of three numbers. Saying it like that makes it easy to generalize to other dimensions. Showing how a list of numbers corresponds to an arrow with one length (the norm) and several angles (depending on the dimension) would then help a lot of kids understand a lot of stuff. There are no physical vs mathematical camps here, physicists and mathematicians share the common language of linear spaces and get along just great. Some of my best friends are physicists. They get to blow things up. Every field has jargon, we can get around that by just saying more about exactly what we mean. Pete St.John (talk) 19:07, 26 February 2008 (UTC)


 * To appreciate the controversy surrounding the 5 different meanings of vector, check out Vectors by Parry Moon (Massachuesetts Institute of Technology) and Domina Eberle Spencer (University of Connecticut), D. Van Nostrand Company, 1965 (see a partial quote above in this talk page). Look at Appendix B Definitions of Scalar and Vector (p. 317 - 322).  There various defintions of scalar (20 defintions from the likes of J.W. Gibbs, O. Heaviside, and M.R. Spiegel) and vector (22 definitions from the likes of J.W. Gibbs and M.R. Spiegel) from 1884 to 1959 are analyzed.  Moreover, even standard calculus books (i.e. ones by Thomas/Finney or ones competing with these two authors) avoid the "ordered list of three numbers" from the linear algebra definition of a "vector".  Thomas also hailing from Massachuesetss Institute of Technology provide a standard voice for mathematicians, a criteria with which to compare your claims about what mathematicians should and do call quantities.  The first page of Chapter 10, Vectors and Analytic Geometry in Space, on page 699 of 8th Edition Calculus Analytic Geometry, Addison-Wesley, 1993, reads "Quantities that have direction as well as magnitude are usually represented by arrows that point in the direction of the action and whose lengths give the magnitude of the action in terms of a suitably chosen unit. ...When we work with arrows in mathematics, we think of them as directed line segments and we call the sets of equivalent segments vectors"  Here in a leading calculus text written by Mathematicans held in the highest regard, quantities are said to belong to the set of arrows and arrows are equatated to vectors.  I have unfailingly provided references and quotes for my definitions, mathematical and physical.  I would appreciate the same courtesy whenever possible. So could PeterStJohn show me the same courtesy and provide quotes from 3 or 4 references (I have provided 4 above from leading physics texts and 1 from a leading calculus text--all categorically definining vectors as quantities) for the defintion of vector that he advocates? --Firefly322 (talk) 01:36, 27 February 2008 (UTC)

Merge
Suggest that the stub vector (physical) be moved here before it gets further forked. It may be a POV fork, rather than a content fork (disallowed), or a contexutal fork (possibly allowed). &mdash; Arthur Rubin | (talk) 22:28, 26 February 2008 (UTC)


 * Maybe userify (sandbox). It's likely an attempt to scratchpad this article, and write it from a point of view that Firefly finds acceptable.  If it stays in the mainspace, it's going to fork away from other articles in a bad way.  Also, nearly everything in the article already has a (quite detailed) Wikipedia article of its own.  However, Firefly should be encouraged to present his/her ideas in a way that can be more naturally merged in with existing articles, and I think a sandbox could be a good way to do that. Silly rabbit (talk) 22:36, 26 February 2008 (UTC)


 * There is nothing to merge at the moment (besides the table of content): it's just the layout of a non-existing article. But if he is interested in presenting all these topics systematically, why not write it as a wikibook, then? Arcfrk (talk) 22:51, 26 February 2008 (UTC)


 * Also I am wondering if Arthur Rubin still insists that a vector is not a quantity? As he did in the discussion page from quantity.  If so, I am extremely uncertain he would understand or appreciate the motivation behind vector (physical). --Firefly322 (talk) 00:46, 27 February 2008 (UTC)


 * What is a wikibook? --Firefly322 (talk) 00:46, 27 February 2008 (UTC)


 * A vector is not a quantity. Whether it's a "quantity" under other circumstances is another matter.
 * A wikibook is another project, which has different rules. &mdash; Arthur Rubin |  (talk) 01:06, 27 February 2008 (UTC)


 * Support - I tend to agree with Arthur Rubin that there should be a merger, although a userfy per Silly Rabbit might be ok if the author could satisfactorily explain the aims of the article. At the moment, I see no justification for the article to be in mainspace. - Neparis (talk) 01:12, 27 February 2008 (UTC)


 * A vector as a physical quantity corresponds to what 1st year calculus and 1st year physics texts present. As shown through quotation of leading texts, the vector of a 1st year calculus or a 1st year physics text is a quantity, while a spatial vector from a vector space/linear algebra text--as granted by Arthur Rubin--is NOT a quantity and that in essence is my motivation and justification for creating vector (physical).   --Firefly322 (talk) 01:56, 27 February 2008 (UTC)


 * No, what this shows is that (some) physical quantities of 1st year physics are vectors. Note the difference, and see a more detailed explanation below. Arcfrk (talk) 01:59, 27 February 2008 (UTC)


 * I still do not see a distinction in terms of the mathematics. - Neparis (talk) 05:15, 27 February 2008 (UTC)


 * Support. Firefly's proposed article on "vector (physical)" as "a quantity with direction conforming to physical laws" makes little sense to me.  First, speaking as a practicing physicist, there are lots of vector spaces and Hilbert spaces in physical sciences that have nothing to do with a "direction".  Second, those "vector" objects that do have a "direction" belong in vector (spatial). —Steven G. Johnson (talk) 05:19, 27 February 2008 (UTC)
 * Mr. Johnson are you saying that a velocity vector, an acceleration vector, and an electric field strength vector, are all spatial vectors? In my mind, mere unit analysis would contradict such an idea. --Firefly322 (talk) 04:01, 28 February 2008 (UTC)

Am I the only one who isn't getting any text in that article (except the first line)?? Firefly322, if you want to write an article but only have headings right now, you should seriously consider using your userspace to develop it at least to have some content. However, there are concerns about duplicating/forking content, so maybe you should be sure it's going to be a productive/worthwhile use of your time. I would also point out that it is not our job to teach first-year physics with calculus, nor are we here to write articles "For Physicists and Engineers." --Cheeser1 (talk) 05:21, 27 February 2008 (UTC)

Note: The text of the article vector (physical) has been copied to a new article vector (Gibbs-Heaviside). Silly rabbit (talk) 02:21, 28 February 2008 (UTC)


 * Right. I did a google search on Gibbs-Heaviside and found that there are several links that use the term.  And I adopted the term from Michael J. Crowe's book A History of Vector Analysis. --Firefly322 (talk) 04:03, 28 February 2008 (UTC)


 * Why are you creating duplicate articles when neither article has any content?? --Cheeser1 (talk) 04:31, 28 February 2008 (UTC)


 * That's an opinon that is not supported by the facts. The article has substantial and significant references, which is the most important thing according to the wikipedia guidlines for creating a new article.  It also has a clear up-front defintion (with links) and a working outline. --Firefly322 (talk) 05:16, 28 February 2008 (UTC)
 * And no content. There is no text in these articles. It has been suggested and agreed by everyone but you basically that this needs to be moved to a sandbox where you can write an article and see. WP:RS doesn't mean if you have some sources at the bottom and nothing else that you have a reasonable article. --Cheeser1 (talk) 05:49, 28 February 2008 (UTC)
 * I think your missing several points of wikipedia guidlines as well as being uncivil. --Firefly322 (talk) 12:37, 28 February 2008 (UTC)

Of vectors and quantities
I'd like to clarify the unfortunate confusion: there are physical quantities that are vectors, just as there are physical quantities that are scalars, tensors, spinors, and so on. This does not mean that, conversely, vectors, scalars, tensors, or spinors are quantities. The word "vector" in the expression "vector quantity" is a modifying adjective, just like a "school bus" is a bus whose purpose has something to do with schools — this doesn't mean that "bus" is a type of a school. Thus "velocity is a vector quantity" means that velocity (as a physical entity) is a quantity, and that this quantity has vector character, namely, that it has magnitude and direction. Before Firefly goes through with his ambitious program of revamping the basics of vectors, may I suggest that he learn what they are? Arcfrk (talk) 01:57, 27 February 2008 (UTC)

Please, I have provided an objective basis of references that show the origin of my definition of vector (see my 5 quotes from Calculus Texts below in addition to the 4 quotes from Physics Texts above). Arcfrk's personally directed comment towards me ( "I suggest that he learn what they are?") seems unwarranted given that neither he nor any other contributor taking his same position have provided a significant sample of references and quotes (actually not a single reference nor a single quote) to back up what is being claimed as the general, scientific defintion of a vector. --Firefly322 (talk) 05:06, 27 February 2008 (UTC)


 * This debate seems a bit pointless to me. The English word "quantity" is sufficiently vague in ordinary usage that few scientists or mathematicians would blink at the term "vector quantity", and a debate over whether all vectors are "quantities" seems an ill-defined venture into linguistics to me.  It is sufficient to define what we mean in this article precisely in mathematical terms without hand-wringing over metaphors.  —Steven G. Johnson (talk) 05:30, 27 February 2008 (UTC)

Definitions of Vectors from a sample of Calculus Texts
1) From Calculus (4th Edition) by Howard Anton c1992, Drexel University, 1992, John Wiley & Sons; Ch.14.2, p.845

"Many physical quantities such as area, length, mass, and temperature are completely described once the magnitude of the quantity is given. Such quantities are called scalars.  Other physical quantities, called vectors, are not completely determined until both a magnitude and a direction are specified. ... In this section we shall develop the basic mathematical properties of vectors."

2) From Calculus: Concepts and Contexts (2nd Ed) by James S Stewart c2001, McMaster University, Brooks/Cole; Ch 9.2, p 652

"The term vector is used by scientists to indicate a quantity (such as displacement or velocity or force) that has both magnitude and direction."

3) From Schaum's Outline of Theory & Problems of Advanced Calculus by Murray R. Spiegel c1962, Rensselaer Polytechnic Institue, Schaum Publishing Company; Ch 7, p. 134

"There are quantities in physics characterized by both magnitude and direction, such as displacement, velocity, force and acceleration. To describe such quantities, we introduce the concept of a vector as a directed line segment PQ from one point P called the initial point to another point Q called the terminal point."

4) From Schaum's Outline Series Theory & Problems of Differential and Integral Calculus (3rd Ed) by Frank Ayers, Jr. and Elliott Mendelson c1990; Ch 23, p. 155

"Quantities such as force, velocity, acceleration, and momentum, which have both magnitude and direction are called vector quantities or vectors."

5) From Multivariable Calculus by William G. McCallum, Deborah Hughes-Hallett, Andrew M. Gleason et al. c2000, Wiley Custom Services; Ch. 12 p. 59

"A Fundamental Tool: Vectors In one-variable calculus we represented quantities such as velocity by numbers.  However, to specify the velocity of a moving object in space, we need to say how fast it is moving and in what direction it is moving.  In this chapter vectors are used to represent quantities that have direction as well as magnitude." --Firefly322 (talk) 05:06, 27 February 2008 (UTC)


 * I'm not sure what point you are trying to make. None of the sources you quote contradict the vector (spatial) article, although being first-year textbooks they are necessarily a bit simplistic.  For a more precise definition (formalizing what is meant by "having a direction"), see e.g. George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001).  —Steven G. Johnson (talk) 05:22, 27 February 2008 (UTC)


 * In math and physics, the usage of quantity can be broken into two categories: a continuous magnitude (with or without direction) or a discrete multitude. The fact that there are only two possible formal meanings makes it an almost perfect word for a wide range of mathematical and physical definitions: the surrounding language in the definition allows the for the context to select either magnitude or multitude or both.  It has just the right amount of intellecutal wiggle-room (not too much and not too little).  Besides, if the word quantities were as vague as Mr. Johnson claims, then why would Afrken use as frequently as he does in this passage from Mathematical Methods for Physicists :


 * Lots of English words are vague, but we still write textbooks in English and use English words. We deal with the vagueness by adding precise definitions when precise definitions are called for.  —Steven G. Johnson (talk) 18:56, 27 February 2008 (UTC)





Mr. Johnson wrote: "None of the sources you quote contradict the vector (spatial) article." Actually, from the way I see it, they do. All sources refer to a vector as a physical quantity. The vector (spatial) article tries to be about vectors that need not obey any physical laws (i.e., the basis vectors [pure mathematical objects] of mathematically defined vector space) and about physical vectors as described in the quotes from the 6 calculus (including Thomas/Finney) texts and 5 physics texts (including Afrken). The problem is that this particular duality can become psychotic (especially when it's not clear that it exists) for Engineers and Scientists. Moon and Spencer don't say psychotic, but obviously they were motivated to write a serious analysis of the differences for their courses on Vectors at MIT and UConn by the introduction of a mathematical vector space terminology sometime around the early 1960's.  --Firefly322 (talk) 07:13, 27 February 2008 (UTC)


 * Sure, spatial vectors are used a lot to describe physical laws. But there's a big difference between saying that and asserting that the only things that are spatial vectors are those described by physical laws.  (Physicists make up unphysical laws all the time to explore ideas, and even for practical purposes such as to define absorbing boundaries in computer simulations.) And this article is explicitly not about elements of any arbitrary vector space: spatial vectors have a specific relationship to spatial coordinate systems.  —Steven G. Johnson (talk) 18:56, 27 February 2008 (UTC)


 * If this article is truly about vectors in a physical spatial coordinate system, then a reader would expect such vectors to obey physical laws. If not, the reader should be told up front that physical laws need not apply. And if so, an entry on the physical vectors conforming to physical laws would be expected. --Firefly322 (talk) 20:42, 27 February 2008 (UTC)


 * What does being in a spatial coordinate system have to do with obeying physical laws? It is perfectly possible to write nonphysical laws that use Cartesian coordinates.    This is like saying that the article about real numbers should state whether physical laws apply.  I think you are philosophically confused.  —Steven G. Johnson (talk) 00:09, 28 February 2008 (UTC)


 * In terms of "vectors" related to (a) physical space, they can either transform as "vectors" or "co-vectors" with respect to coordinate transformations, or they can have dimensions of units of space (times other units) or reciprocal units of space, or some other combination. None of the modifications deserve an article.  First year calculus texts do not necessarily contain useful or correct information, even if "reliable".  &mdash; Arthur Rubin |  (talk) 12:47, 27 February 2008 (UTC)


 * I completely disagree with the statement "First year calculus texts do not necessarily contain useful or correct information, even if "reliable". ". First year calculus books are an introduction to Gibbs-Heaviside Vector Analysis.   Maxwell's equations were formulated in Gibbs-Heaviside Vector Analysis and their success in this form in terms of results and users (whereas in Quaternion form, there were few results and few users) is what made Afrken in Mathematical Methods for Physicists write:

"The need for vector analysis became apparent only with the development of Maxwell's electromagnetic theory and in appreciation of the inherent vector nature of quantites such as the electric field and magnetic field." Besides, what relevance is a linear algebra term like "covectors" to a discussion on the value of the Gibbs-Heaviside vectorial analysis system? --Firefly322 (talk) 18:42, 27 February 2008 (UTC)


 * First year calculus books are an introduction to Gibbs-Heaviside Vector Analysis. No, first-year calculus books are an introduction to calculus. Most calculus curricula don't even touch vectors until the third semester. Furthermore, Wikipedia is NOT a textbook. We should not be presenting information in a particular style or with particular pedagogical concerns. --Cheeser1 (talk) 19:11, 27 February 2008 (UTC)


 * A spatial vector, a "mathematical entity" or "mathematical object," from a vector space implies advanced mathematical "pedagogical concerns." For any math or physics wikipedia article, I believe that the multi-disciplinary audience of a 1st year calculus book and physics book is a much better target than the 2nd year mathematical audience of a linear algebra book where spatial vectors are introduced. The relevant point is that an introductory calculus book is coherent.  The meaning of the variables in such a book are carefully selected so that they are scientifically relevant to any future scientist or engineer, i.e., meaning of these variables are in general Gallian-Newtonian for non-vector functions and Faradian-Maxwellian for vector functions. As the quotes from the standard calculus texts show vector as a quantity is used (and further quotes would show that quantity is paired with the usage of the word particle, which together imply physical fields such as physical space and physical time).  In first-year calculus books, the calculus operations (differentiation and integration) on Gibbs-Heaviside vector function are implicitly defined over a Maxwell-Faraday Field (not in or over an abstract Hilbert-like vector space).  Faraday, given his start in a book store and the fact that humanties majors can and do read him in the orginal since he almost never used any mathematics, is exactly the kind of core audience that Wikipedia is written for.   --Firefly322 (talk) 19:46, 27 February 2008 (UTC)

Common ground should exist on Precision and accuracy
(copied from above for clarity)

Lots of English words are vague, but we still write textbooks in English and use English words. We deal with the vagueness by adding precise definitions when precise definitions are called for. —Steven G. Johnson (talk) 18:56, 27 February 2008 (UTC)

Precision and accuracy is exactly what I'm focused on. Though I don't at all share your idea that quantity or any other common--yet formally used words--from a standard calculus text are as vague as you seem to maintain. For example, I maintain that the definition I have crafted for a physical vector is very precise and very accurate:

--Firefly322 (talk) 20:34, 27 February 2008 (UTC)
 * Vector (physical), a quantity with direction conforming to physical laws such as quantities of displacement, velocity, and acceleration. (for Physicists and Engineers)


 * The wording seems more precise than accurate, but perhaps is a starting point for our mutually understanding the differences. "Vector [is] a quantity with direction" sounds like the quantity has a direction; like, "3 is pointed north" but we mean "the wind is from the south at 3 mph", the vector has magnitude 3 and orientation northwards. "Vectors are abstract structures of quantities which satisfy the rules of vector spaces. Physical vectors have quantities interpreted as magnitude and direction, which can be measured from physical processes, such as wind where magnitude reflects the speed of the wind and direction is compass direction on the surface of the Earth." Pete St.John (talk) 23:15, 27 February 2008 (UTC)


 * Calling them "physical" vectors is a misnomer, as there are lots of vector spaces in physics that aren't associated with "directions" per se. (e.g. the charge density is in a vector space of real-valued functions.)  That's why the article is titled "vector (spatial)": it is only those vector spaces that have a particular relationship to spatial coordinates (i.e., a "direction", although this can be defined more precisely in the language of differential geometry or by describing how the components transform under rotations, as in the last section of our article).


 * I really don't understand why people are getting so hung up on the word "quantity". If you google "vector quantity" (with quotes), you get almost 100,000 hits: in common English usage, the term "quantity" has become broadened to include things that are not simply real numbers.  The Oxford English Dictionary lists one of the meanings of "quantity" as "the property of things that is in principle measurable."  Can we stop this silly argument over whether a vector is a "quantity," please?  It has nothing to do with the precise mathematical definition of a "vector" (spatial or otherwise), nor with the title of this article.


 * —Steven G. Johnson (talk) 23:55, 27 February 2008 (UTC)
 * I used "physical" in response to the user of that word in the proposed defintion given by Firefly. I believe some of the editors want to distinguish a class of vectors (that would include momentum) from another class of vectors (maybe Hilbert spaces in the case that the dimension is not finite, for example). If somebody would list some examples of vectors that are "spatial" and some that are not, we might be able to figure out the classification. To mathematicians, theoretical physicists, electrical engineers (lot's of people who use math beyond the level of calculus), the distinction of "spatial" vectors from "vectors" is not obvious. It is certainly not obvious to me, but I guessed at an interpretation hoping that it could lead to mutual understanding eventually.
 * Regarding "quantity"; the definintion of vector explicitly distinguishes vectors from scalars. Scalars are definitely quantities; in fact they are essentially a formal mathematical definition for the notion. Vectors are quantitative, but they are not single quantities; they are structures of quantities. Imagine an arrow with a length and a direction. An arrow is not a number, but it's a fine model of a vector. An actual physical arrow, that you could shoot from a bow, is a great model of a vector in three-dimensional euclidean space. Pete St.John (talk) 00:05, 28 February 2008 (UTC)


 * Yes, this article does distinguish a class of vectors from arbitrary vector spaces. There is a precise definition given in the last section of the article of what is meant by "having a direction":  a spatial vector's components must transform like the coordinates under rotations.   These could also be termed "rank-1 contravariant tensors" and are also called "polar vectors", but both of those terminologies would be forbidding to neophytes.  Under this definition, things like the momentum are vectors, but things like charge density functions are not.  It has nothing to do with finite-dimensional vs. infinite-dimensional, nor with "physical" versus "unphysical."


 * I agree that this article has been the subject of persistent confusion by editors who don't understand the distinction between general vector spaces and contravariant vectors, but the subject is perfectly well defined and totally standard (see e.g. Arfken and Weber, Mathematical Methods for Physicists). There is no point to having this article (separate from vector space) if we don't restrict it to this particular class of vector spaces.


 * (In an earlier version of this article, the precise definition came earlier rather than being exiled to the last section. The various shufflings and rearrangements that this article has undergone have, in my opinion, made it less clear what, precisely, it is about.)


 * I'm not going to get into a silly debate over whether a vector is a quantity, except to say that usage is king when it comes to language, and the term "vector quantity" is widespread. No one here needs a condescending lecture that a vector is not a single real number (except in one dimension).


 * —Steven G. Johnson (talk) 00:20, 28 February 2008 (UTC)


 * (EC) In light of the below, I'll make this short. I don't know what "spatial vector", the title of this article, is supposed to mean. If the definition is "rank 1 contravariant tensors" then we have a different, but more technical, problem to address. Do all the editors here, save one, agree on a defintion of "Vector (spatial)" for the purpose of this article? Pete St.John (talk)

Stop the trolling
According to the latest statistics, this talk page is now 75K long, and without a iota of improvement for the article. It seems clear to me that one user (Firefly, who also occasionally uses an anon account) came to this page with his unorthodox personal opinions about vectors (see his comments about them being "controversial", or the insistence on calling them "Gibbs-Heaviside vectors") which are not supported by the community, and tied up a lot of experienced and even expert editors. As always, if there is no real substance to discuss, the debate tends to degenerate into interpretations of "how said what" type. Thus, a plea to all the participants:


 * Do not feed the troll.

I believe that we have done more than enough to address Firefly's concerns. It is not the purpose of wikipedia to educate potential editors. Talk pages exist to discuss improvements to the articles, not as a general forum for discussions (this is an official wikipedia policy). What is happening here is a disruption to wikipedia's normal editing process. If the disruption continues, it should be handled in the administrative way. I also suggest archiving the recent long and unproductive discussion. Arcfrk (talk) 00:51, 28 February 2008 (UTC)


 * The requests for attention at wikiproj Math range from 7 days old to ...today. I think that it will take a bit of time to acquire that first iota of improvement mentioned. My hope has been to learn what definitions are intended so that the article can conform to a definition. Right now I have no clue what is meant, because I'm not familiar with the distinction. I've always thought that physicists and mathematicians meant the same thing by "vector", and I suspect that generally we do. Pete St.John (talk) 01:17, 28 February 2008 (UTC)

Origin of Gibbs-Heaviside Vector Terminology

 * The reference--as the primary guideline of wikipedia calls for--justifying my usage of the Gibbs-Heaviside vector comes from Michael J. Crowe's A History of Vector Analysis, Dover, 1992 (see link below). This book received the very prestigious Jean Scott Prize by the Maison des Sciences de l'Homme (Paris)in 1992.  Crowe coined the term "Gibbs-Heaviside" vector to distinguish it from other types of vectors.


 * To be called a troll in light of the quality of references I have provided is a reprehensible comment and Afrck should be warned against such personally demeaning and unwarranted wording in the future. For I have spent countless hours carefully reading a substantial set of books ranging from Gibbs to Crowe.

http://books.google.com/books?id=g0T8hPo7i-4C&pg=PR3&lpg=PR3&dq=jean+scott+prize+by+the+maison+des+sciences+de+%22l+homme%22+paris+in+1992&source=web&ots=ViNbNqHV2k&sig=XIQeEA-xT1xzxZdy44eeCo_VVCo&hl=en

I see much common ground with Mr. Steven G. Johnson (I think most reasonable contributors will also). I propose that much of this text be moved to an archive and that we continue the dicussion in a way that is as civilized and as open-minded as humanely possible. --Firefly322 (talk) 01:45, 28 February 2008 (UTC)


 * If there are reputable sources clearly referring to spatial/contravariant vectors as "Gibbs-Heaviside" vectors, the article should mention this terminology. However, Wikipedia policy is to name articles using the most common name, and most often these things are just called "vectors" (although we have to add a parenthetical disambiguation); explicit mentions of Gibbs and Heaviside are extremely rare nowadays.  However, a Google search for "Gibbs-Heaviside vector" turns up no usages of that term as a noun as far as I can tell; all the links I've followed seem to use "Gibbs-Heaviside vector calculus" and "Gibbs-Heaviside vector analysis" etcetera.  Are you sure you're not misreading your sources?  One also has to be careful as to what, precisely, is being referred to, as there are many types of things called "vectors" nowadays.  In any case, tracking down references for obscure synonyms seems to be the least of the concerns about this article right now.  —Steven G. Johnson (talk) 06:02, 28 February 2008 (UTC)
 * The original usage appears to be (from page 262) "Gibbs-Heaviside system of vector analysis: number of books on, before 1911, 225-226" (see link above for these pages) The above link may ommit these pages unfortunately. --Firefly322 (talk) 12:27, 28 February 2008 (UTC)

Discusion on the term Physical Vector
(Copied from above for clarity)

Calling them "physical" vectors is a misnomer, as there are lots of vector spaces in physics that aren't associated with "directions" per se. (e.g. the charge density is in a vector space of real-valued functions.) That's why the article is titled "vector (spatial)": it is only those vector spaces that have a particular relationship to spatial coordinates (i.e., a "direction", although this can be defined more precisely in the language of differential geometry or by describing how the components transform under rotations, as in the last section of our article). I really don't understand why people are getting so hung up on the word "quantity". —Steven G. Johnson (talk) 23:55, 27 February 2008 (UTC)

By physical vector I mean that a real-world device can be constructed to measure and verify the existance of such a quantity. The focus is on its measurability and not on its spacial coordinates. But I'm open to what you are saying. I find much of what you write persuasive. The function of any scalar quantity like charge density that is distributed over space would be defined with position vectors, but this would be a scalar function of spatial vectors. I don't think I'm philosophically unsound. An article simply about spatial vectors would be great, but other vectors like displacement (which is conceptually different than a position or spatial vector), velocity, acceleration, and electric field strength have as much merit and value to Wikipedia. Based on unit analysis, all these types of vectors need a different name it seems than just spatial. --Firefly322 (talk) 02:29, 28 February 2008 (UTC)

Vector (physical), a quantity with direction conforming to physical laws such as quantities of displacement, velocity, and acceleration. (for Physicists and Engineers) --Firefly322 (talk) 20:34, 27 February 2008 (UTC)


 * The wording seems more precise than accurate, but perhaps is a starting point for our mutually understanding the differences. "Vector [is] a quantity with direction" sounds like the quantity has a direction; like, "3 is pointed north" but we mean "the wind is from the south at 3 mph", the vector has magnitude 3 and orientation northwards. "Vectors are abstract structures of quantities which satisfy the rules of vector spaces. Physical vectors have quantities interpreted as magnitude and direction, which can be measured from physical processes, such as wind where magnitude reflects the speed of the wind and direction is compass direction on the surface of the Earth." Pete St.John (talk) 23:15, 27 February 2008 (UTC)


 * Let me try to explain a quantity with direction better. We often see Newton's Latin Principia's laws of motion translated to English with the phrase "quantity of momentum."  Such a quantity with direction can be transferred to another mass under the simple rules of 3-dimensional vectorial addition and subtraction.  The transfer to another mass will conserve the quantity of momentum and transform its direction. --Firefly322 (talk) 03:05, 28 February 2008 (UTC)


 * Three hopefully uncontroversial facts:
 * Historically, the term "quantity" referred to things that could be measured by a single number, and this is still probably the most common usage.
 * Nowadays, many many people (google it) talk about a "vector quantity" and "vector quantities", hence evolving the meaning/usage of word "quantity" in this context.
 * Whether one uses the word "quantity" to describe a vector is irrelevant to the precise mathematical definition.
 * What is the point of this debate over the meaning of "quantity"? How does it affect what should or should not go into the article?


 * —Steven G. Johnson (talk) 06:12, 28 February 2008 (UTC)


 * Believe me, I would like for us to have three uncontroversial facts agreed to. Maybe we do, but not these (except perhaps that vector quantities is good, meaningful diction).
 * Alas googling "'quantity of momentum' Newton" turned up this passage:
 * Newton defines momentum as follows: "The quantity of momentum is the measure of the same, arising from the velocity and quantity of matter conjointly." (http://www.utmost-way.com/issacnewtonarticle.htm)
 * So Newton used the word quantity in what I would say is a vectorial sense.
 * Nowadays, the mathematical entity of a vector overshadows its original meaning as a physical description of many kinds of energy (mechanical and electrical).
 * The debate over quantity centered on these two meanings of vector:
 * Nature side of debate: A physical force is a quantity and this quantity can be called a vector (which is still the main meaning of vector according to www.m-w.com)


 * comment Momentum is generally considered a vector, and momentum is not a Force (as defined by Newton); so to imply that vectors are forces is unclear language. Pete St.John (talk) 21:31, 28 February 2008 (UTC)
 * From this point of view, Physical Vector is accurate (a quantity of a vector has meaning from this usage and viewpoint)
 * Mathematical side of debate: A mathematical symbol also called a vector can be used to represent a quantity of physical force (here calling a mathematical symbol a quantity would be ludicruous if the symbol and the quantity didn't happen to share the same name as they do)
 * From this point of view, Mathematical Vector is accurate --Firefly322 (talk) 07:56, 28 February 2008 (UTC)

Regarding your other statements above: displacement, velocity, acceleration, and electric field strength are spatial vectors (in the precise sense that they transform like the coordinates under rotations, or informally in that they have a "direction" in space). The definition of a spatial vector is based on contravariance, and has nothing to do with units per se. We don't need another article just to say that many spatial vectors are described by (or are used to describe) physical laws. —Steven G. Johnson (talk) 06:16, 28 February 2008 (UTC)
 * Actually, now in light of the two very, very different meanings of vector (one symbolic conforming to mathematical law, the other natural conforming to physical law), I would say proceeding in our discussion with extra caution is best. I think we can easily find several points of agreement and common ground once we figure where we stand in agreement and disagreement on the meanings of vector and the meanings of quantity.   --Firefly322 (talk) 08:07, 28 February 2008 (UTC)
 * Which begs the question: are you interested in other people's input, or are you going to fill this talk page with your ideas about vectors, citing things like, say, Newton, which was written centuries ago and reflects neither the current use of the English language, nor the current standards of mathematics or physics? --Cheeser1 (talk) 08:23, 28 February 2008 (UTC)
 * To disregard Newton is something even the likes of Einstein would not do. --Firefly322 (talk) 12:16, 28 February 2008 (UTC)
 * I had a seance today and, actually, Newton and Einstein agree with me. —Steven G. Johnson (talk) 16:38, 28 February 2008 (UTC)
 * To amplify on the previous sarcastic comment, it would be fatuous to assume that Newton used the term "quantity" (or "vector") in the same sense that we do today. &mdash; Arthur Rubin |  (talk) 18:35, 28 February 2008 (UTC)