Talk:Cross product/Archive 1

Cross product in N dimensions
NOTE: This section contains comments which were originally posted separately at different times about the generalization of cross product in N dimensions

7 dimensions
I removed the following from the 7-dimensional cross product: This isn't true in 3 dimensions, so I doubt it's true in 7. Note that x&middot;y is not the cosine of the angle between x and y. AxelBoldt 02:06 Apr 30, 2003 (UTC)
 * |x&times;y|2 = |x|2|y|2(1-(x&middot;y)2)

I added the correct formula. AxelBoldt 15:56 Apr 30, 2003 (UTC)

3+1 and 7+1 are powers of 2
If you can do cross products for 3D and 7D vectors using 4D and 8D techniques, what about other Ds? Can you do 4D cross products using 5D or is there something special about 3D and 7D (like 3+1 and 7+1 are powers of 2 or something)? -- SGBailey 22:05, 2003 Nov 16 (UTC)


 * Yes, 3 and 7 are special. It has to do with the fact that the only finite-dimensional real division algebras have dimensions 1, 2, 4 and 8, and the first two only give trivial cross products (anything times anything is zero). You are right to think that there is something significant about these numbers being powers of 2. --Zundark 22:46, 16 Nov 2003 (UTC)

A possible generalization to N dimensions
I am not sure about this, but it seems that if the cross product in 3 dimensions needs 2 vectors to produce an orthogonal vector, then this can be extended to 2 dimensions by needing only 1 vector (rotating it 90 degrees) to again produce an orthogonal vector. The same can be done in 4 dimensions with 3 vectors and so on. - Lmov 06:36, 20 Jan 2004 (UTC)


 * Yes, that works. Just use the determinant definition of the 3-dimensional cross product, and extend it in the obvious way. --Zundark 13:22, 20 Jan 2004 (UTC)


 * Although it's a fine method of obtaining orthogonal vectors, it doesn't preserve the other nice algebraic properties of the standard (2^n)-1 cross product... Conskeptical 12:27, 27 Apr 2005 (UTC)
 * Yay for timely responses too... oh well. I'll drop this newbie style soon enough i hope... Conskeptical 12:28, 27 Apr 2005 (UTC)

Crossproduct dimension
I added the words "in a three dimensional" (vector space) to the first paragraph. A cross product can only be a binary operation in R3. It is a unary operation in R2, and a trinary operation in R4. If anyone wants to go through the article and separate R3 versus general cross product facts, I welcome it. (I actually came to this page hoping for the determinant form of a R4 cross-product of 3 R4 vectors!) Tom Ruen 23:20, 18 September 2005 (UTC)
 * Adding a bit of clarification in the first paragraph is no problem with me. Since the 3D crossproduct is by far the most used form of the cross-product, I find it very natural that the first part of the article and most of the material in the article is dedicated to it. If you wish to improve upon the other dimension generalizations of the cross-product (see sections at the bottom), you are more than welcome. Oleg Alexandrov 23:40, 18 September 2005 (UTC)
 * Feeling adventurous, I added a paragraph about n-ary cross products in Rn+1. They're simple enough to describe, but I've never formally learned them from any course or text, so I'm not sure if there is a standard notation.  Furthermore, how do you describe the n-dmensional analogue of the sentence "the volume of the parallepiped"?  I said "hypervolume bounded by some vectors", but that really doesn't sound right to me. -Lethe | Talk 18:45, 10 November 2005 (UTC)

The two-dimensional cross product
It's relatively common (especially when solving the Euler equations in two dimensions) to define a cross product of two-dimensional vectors by extending them with $$0\hat z$$, taking the normal cross product (necessarily yielding $$k\hat z$$, where k may be 0), and then calling just k the product (since the $$\hat z$$ obviously is useless). The result is quite useful &mdash; it's still anticommutative, and still measures area, for instance. I see no mention of this in the article; is there another name for it I don't know, or is there some reason it doesn't belong, or what? --Tardis 14:02, 16 January 2007 (UTC)

"The cross product is not defined except in three-dimensions"
I'm dead certain this is wrong; logically the cross product would only need a minimum of two dimensions and take n-1 vectors, where n is the number of dimensions. (Or that is to say, that's what I was told.)

124.191.99.134 08:32, 29 July 2007 (UTC)


 * The cross product generalizes to an (n-1)-ary operation in n dimensions, but this is not usually called a cross product when n is not 3. The article already covers this in the section Higher dimensions. --Zundark 09:09, 29 July 2007 (UTC)

My formula vs yours
Am I missing something, or is this wrong?



\mathbf{i}(a_2b_3) + \mathbf{j}(a_3b_1) + \mathbf{k}(a_1b_2) - \mathbf{i}(a_3b_2) - \mathbf{j}(a_1b_3) - \mathbf{k}(a_2b_1) $$

Either it is incorrect or somehow misleading. I am no math person, I am merely a highschool student with a final tomorrow, and I checked wikipedia to see if i was rigth about how to do cross products. The way I learned, and the way verified by a cross-product calculator i found online, is



\mathbf{i}(a_2b_3-a_3b_2) - \mathbf{j}(a_3b_1-a_1b_3) + \mathbf{k}(a_1b_2-a_2b_1) $$

I know now for a fact the above works, and as far as I can tell it is not equivalent to the first on (note the double negative in the j term). I may very well be missing something, as it's late and I'm stressed, but if this is wrong it should be fixed. Even if they are equivalent, I think my way is more succinct and easier to understand. Thanks, Personman 05:48, 20 May 2005 (UTC)


 * Your expression is wrong, since it should have +j rather than -j. You can see this by expanding the determinant form. --Zundark 12:04, 20 May 2005 (UTC)

Im an engineering student and am going to have to agree with the first poster. You subtract the determinant of j-hat. dont know why. take a gander at chapter 4.3 of edition 11 of engineering statics and dynamics by hibbelier. dont know in what country your in that u do this in high school 216.197.255.21 03:22, 21 October 2007 (UTC)


 * You are wrong. I doubt that Hibbeler is wrong (though typos in text books are not unknown), so you might like to read what he says again, carefully noting the sign of all terms, particularly the ones that j-hat is being multiplied by. Perhaps you should also contemplate the meaningfulness of the phrase "determinant of j-hat". (By the way, I'm in England, as clicking on my link would have shown. We don't have high schools.) --Zundark 07:54, 21 October 2007 (UTC)

k i don't no how to do your fancy math notation on wikipedia. have you taken any enginerring classes or physics classes? the cross product is calculated by taking the minors of i,j,k in the matrix where the first row are i,j,k variables and the second row is composed of values for radius in cartesian vector notation, the third row is force in cartesian vector notation. for minors you must alternate + - + - + -(this is true, must be for matrix inversion) therfore the j-hat minor is negative. leave ur email on my talk page or something and ill email you a scanned copy of the page in question. i might talk a while getting back to you though. also i have prof who have stated wikipedia is wrong with other concepts related to physics, so im trying to clean this up, thats why im doing this. —Preceding unsigned comment added by 216.197.255.21 (talk) 03:49, 31 October 2007 (UTC) Also you could check the notes on determinants for my GE 111 class here. http://engrwww.usask.ca/classes/GE/111/ —Preceding unsigned comment added by 216.197.255.21 (talk) 00:29, 2 November 2007 (UTC)

Edits concerning parallelipiped and Lagrange's formula
I think the stuff about Lagrange's formula is off-topic for this page - it should be linked to, not put on this page. Same goes for the parallel piped. I edited these things previously - but I was very careless in my edit. I now put in a correct link to Lagrange's formula, and i'll put in a link to the triple product now. I'll wait a couple days for comments before I do a less careless cut of this page. Sorry for screwing it up the first time. Fresheneesz 21:48, 9 November 2005 (UTC)
 * I disagree. Those formulas are not off-topic, they show properties of the cross-product. I think moving that Lagrange's formula subsection at the bottom of that section would be a good idea though, as it is a bit more peripherical than everything else in the section. You could also trim it a bit, and refer to Lagrange's formula for details. And I belive the parallelipiped formula is fine where it is, it is an important property. Oleg Alexandrov (talk) 22:01, 9 November 2005 (UTC)
 * It's definately true that the both Lagrange's Formula and the parallelpiped formula are important - however, they're not properties of a cross-product. Lagrange's formula involves gradients and dot products - but I didn't even see the formula mentioned on the pages for those. It is a distinctly separate - important but separate - topic. The parallel piped does also contain a cross product in its formula - but why does this warrent its existance on the page for the cross product. When someone looks up the cross product, do you think it would be more useful for them to see properties of the cross product, or do you think it would be more useful for the definition |a X b| as the area of a parallelogram to be hidden in-paragraph and the equation for the triple product (which has its own page) to be boldly displayed in LaTex on its own line? I'm thinking of usefulness here, not the amount of important information contained on this page. Fresheneesz 02:24, 10 November 2005 (UTC)

I do believe that the volume of the parallelipiped is a very important property and geometric illustration of the cross-product. So, I would like to have it stay. I even changed my mind about Lagrange's formula, I like it where it is. I perfectly agree with you that too much information does not make for a better article, however, in this case things are nicely arranged in sections, the formulas presented are very relevant, the article is rather short and well-organized, so I like it the way it is. Oleg Alexandrov (talk) 04:02, 10 November 2005 (UTC)
 * as long as noone else protests. But I do think that at least putting Lagrange's formula lower or lowest on the page would help things - just because most of the rest of the article pertains to ONLY the cross product relation, and not compound relations. I also will make the expression |a X b| as the area of a parrallelagram larger and more obvious if you want to keep the triple product on this page as well. Fresheneesz 21:05, 10 November 2005 (UTC)

What if you make a new subsection at the bottom of the "Properties" section titled "Identities involving the cross-product" and put Lagrange's formula and related in there? I would like however to keep the volume of the parallelipiped thing where it is, as it is very important, even if it has a dot product in it besides the cross-product. Is that a compromise? :) Oleg Alexandrov (talk) 01:07, 11 November 2005 (UTC)
 * That sounds like a good compromise, i'll do that right now. Fresheneesz 22:50, 11 November 2005 (UTC)

Explaination of the right hand rule
I belive that the current explaination of the right hand rule is ambiguous. I can point the fore and middle fingers in the correct directions in two different ways (yes, it is slightly uncomfortable to do it the wrong way, but that isn't stated)

On the other hand (sorry about the pun), there is only one way to align your straightened fingers with the first vector and then bend them towards the second (unless you are double jointed, but I don't think we need to mention that) --noah 04:02, 10 December 2005 (UTC)
 * Your explanation says:
 * If the coordinate system is right-handed, one simply points straightened fingers in the direction of the first operand and then bends the fingers in the direction of the second operand. Then, the resultant is the vector coming out of the thumb.
 * Well, which are the straightened fingers? As far as I know, all fingers can be straightened. Which fingers get bent? I would say that your explanation is very ambiguous. The present explanation is very clear. It talks about the forefinger, which is only one on each hand. It talks about the middle finger, which there is only one too. Everything is very clear, and this is the classical explanation. You are attempting something fancy which does not help explain things. Oleg Alexandrov (talk) 16:34, 10 December 2005 (UTC)

I learned to do the right-hand rule with the three perpendicular fingers when I was a pup, and I've always taught it that way, and it's been presented that way in most of my books. I can attest to the fact that sometimes my students awkwardly try to switch the middle and index fingers. I usually tell them to make sure they're not flipping anyone the bird. Anyway, one semester, I somehow got stuck teaching a sort of gen. ed. physics course which used a much more remedial text (non-calculus. I don't recall the author), and that book presented it differently: make a flat palm with your right hand, point your thumb in the direction of the first vector, your other fingers in the direction of the second vector, and the resultant vector should point out of your palm. I think this was definitely easier for the kids. But I don't care enough about this stuff to change it, so you can take it or leave it. -lethe talk 13:57, 14 December 2005 (UTC)

What bothers me is that the picture on the cross product page uses different finger assignment for A, B, and A x B than the picture on the right hand rule page. The right hand rule page does state that "Other (equivalent) finger assignments are possible", but since the whole point of the right hand rule is to establish a memorable, useful convention, it would be nice if they agreed. —Preceding unsigned comment added by 137.229.17.63 (talk) 21:26, 13 August 2009 (UTC)

Simple formula
I have read and reread this. Is there a possibility to have an English explanation of cross product? Mainly, an example. Given two XY coordinates, show how the cross product is calculated. In my opinion, this is Wikipedia, not Mathipedia. We shouldn't need a math degree to read it. --Kainaw (talk) 20:13, 31 January 2006 (UTC)
 * Does Cross product help? Oleg Alexandrov (talk) 02:36, 1 February 2006 (UTC)


 * Or the formula
 * a &times; b = [a2b3 &minus; a3b2, a3b1 &minus; a1b3, a1b2 &minus; a2b1].
 * in the section Cross product. Oleg Alexandrov (talk) 02:38, 1 February 2006 (UTC)
 * How do I get to Mathipedia from here? Sounds like a nice place. :-) -lethe talk [ +] 03:38, 1 February 2006 (UTC)


 * While I understand it, it assumes the reader understands that a is a set containing a1, a2, a3... However, if it were written as:
 * (a1, a2) X (b1, b2) = a2 * b1 - b2 * a1
 * then, it is more obvious what is taking place. Or, there can be an explanation of what a and b are.  It isn't obvious to a person who has no clue what a cross product is that a and b are sets. --Kainaw (talk) 23:57, 1 February 2006 (UTC)
 * The problem is that a and b are not sets. They're vectors.  Yes, it happens that vectors can be represented as ordered triples (because they have to be ordered, those aren't properly called sets, but whatever, I take your meaning).  But that representation is not unique, and it's not really consistent to assume that.  I guess we could decide to only talk about the cross product of ordered triples, but that represents a loss of generality.  Every math article attempts to strike a balance between accessibility and comprehensiveness.  It's hard.


 * But you know something else? I don't think you need to know that a can be represented as a triple to understand the very first definition, which is that the cross product is the vector perpendicular to both its multiplicands.  What could be easier?  If the user isn't knowledgeable enough to understand the formal definition, that's OK, because we start with a geometric definition!  Of course, if the reader wants to get all the way through to the bottom of the article, I don't think it's unreasonable to ask that he or she have a familiarity with vector spaces. -lethe talk [ +] 00:18, 2 February 2006 (UTC)


 * I agree that this article could do some with work to make it simple to someone who just wants to know how to calculate the cross product. The section under 'Matrix Notation' partially does this, but I think it would be useful to have a section called something like 'calculating the cross product' which does an example and the general formula, in both column vector and i,j,k styles. This would make the article much more useful. guiltyspark 13:33, 21 November 2006 (UTC)

Can someone please rewrite some of the formulas they appear as latex coding? i will do it later when I have some time otherwise.62.56.27.145 10:11, 14 April 2006 (UTC)
 * I think this is a tempoary network problem see Wikipedia:Village pump (technical)#Network_problem --Salix alba (talk) 11:12, 14 April 2006 (UTC)

Simplest examples
The new "Simplest Examples" subsection is currently part of the "Properties" section. But technically, an example isn't a property.

Perhaps "Simplest Examples" should be its own section, between "Definition" and "Properties"? Or appended onto the "Matrix notation" subsection?

JEBrown87544 23:19, 11 July 2006 (UTC)

Disambiguation: cross products in topology
maybe there should be a remark in the article that the name "cross product" is also used in algebraic topology for various other concepts, see e.g. http://www.win.tue.nl/~aeb/at/algtop-8.html

MathML versus HTML
Shouldn't we use MathML instead of HTML for formulas? --Ysangkok 17:17, 10 January 2007 (UTC)
 * I guess you mean "LaTeX" rather than "MathML". Either way, see math style manual, html formulas are fine. Oleg Alexandrov (talk) 03:40, 11 January 2007 (UTC)
 * We have Internet Explorer 6 installed at work and the LaTeX markup images are coming in in a variety of different sizes in an apparently random fashion. Matthew Townsend 23:23, 9 September 2007 (UTC)
 * Looking again with Safari, the n-hat characters still appear out of place inlined in the text like that. Matthew Townsend 17:53, 12 September 2007 (UTC)

Pseudovectors
NOTE: This discussion was copied from User talk:Edgerck/archive1 as it is relevant to this article.

The result of a vector cross product between two vectors is a pseudovector, not a (true) vector. This is an important difference in mathematics and physics.

I cleaned up the previous definition, by explicitly introducing a right-handed coordinate system that allows the cross-product to be simply (but correctly) defined as a vector. Afterwards, I added a handedness discussion with a cross-product multiplication table for vectors and pseudovectors.

Complications arise here because the cross product is the three dimensional example of what is really a second order tensor. Gibbs is often made to be a villain in math literature for promoting such a cross-creature. However, the usefulness of the cross product -- when correctly used -- is that it makes life simpler for simple things. The flip side is that, up to a certain level, students are not motivated to grasp finer things that might be useful even if just for intellectual growth. Vector analysis (per Gibbs) is presented too often as the "end-all, be-all" of directed quantities. Edgerck


 * Hi. I disagree with your changes to Cross product where you inserted the distinction between vectors and pseudo-vectors. For all practical purposes, I think the cross product of two vectors is a vector. While I believe you are correct in saying that things are more subtle, inserting the distinction between vectors and pseudovectors in many places in the article makes the article harder to read and somewhat confusing I think. I suggest we go back to the previous version, where pseudovectors were mentioned just once, and then one can visit the pseudovector article for more details. What do you think? Thanks. You can reply here. Oleg Alexandrov (talk) 03:18, 5 May 2007 (UTC)

Thanks for your feedback, Alexandrov. Your message is timely as I am just working on a small change to reduce the apparent confusion and motivate the importance of the difference. The result of vector cross product between two vectors is (as we both know) a pseudovector, not a vector. This is an important difference and not only for consistency in wikipedia but also for mathematics and physics. Would the article be less confusing if it would not say what is correct? Possibly at first sight but soon problems would appear as the reader would unwittingly apply that incorrect knowledge. Please continue to watch the page and tell me what you think, after my next change. I will post here when I am done, so you know it's not interim.

BTW, a similar "problem" also appears in the scalar triple product in Triple product -- and there you see that wikipedia already correctly defined the result to be a pseudoscalar. Reading the item below you will find the Vector triple product -- where again the vector product must be correctly used in terms of the rules given in cross product. Edgerck 23:24, 5 May 2007 (UTC)


 * I'd like to note that the edit I mentioned above removed a very important text, which was dealing with the direction of the cross-product (we all know that there exists two different directions perpendicular to a plane). I had not seen your comment above in time, by the way, and I took the liberty of doing a partial revert (partially to restore the text dealing with the direction of a pseudo-vector).


 * My suggestion is that pseudovectors must be mentioned, but not excessively. The primary place where this should be dealt with is in the article about pseudovectors, here just a remark that the cross product can be a pseudovector is enough, I think.


 * Anyway, I will watch this page. Feel free to make changes; my only point is that talking too much about pseudovectors in the elementary cross product articles can make it harder to read. Oleg Alexandrov (talk) 00:04, 6 May 2007 (UTC)

Alexandrov: That statement you reinserted is false. Nature is not limited by our present-day mathematics! The real reason is another, which I will add in my revision. Edgerck 00:10, 6 May 2007 (UTC)


 * What I wrote is not false. You don't need to know anything about pseudo-vectors to talk about cross-product. If you want your text back, please do a good job at writing it well and understandably. There is no point in writing something which is most general yet very hard to understand. Even better, most of that text belongs either in a separate section or in its own article.


 * Cross-product is an elementary article, it is not good to make it too complicated. Oleg Alexandrov (talk) 00:24, 6 May 2007 (UTC)

I was referring to this text: "Fortunately, in nature, measurable quantities involve pairs of cross products, so that the “handedness” of the coordinate system is undone by a second cross product, and the measurement doesn't depend on an arbitrary choice of coordinates." This text should not be there, but I understand the queasiness behind it. Edgerck 00:57, 6 May 2007 (UTC)

Done editing cross product. Edgerck 01:07, 6 May 2007 (UTC)


 * Ah, that "Fortunately... " text. I don't mind that it is cut out.


 * I renamed the section you added to "Cross product and handedness" and I moved it further down. I understand it is very important, but the fact is that it is rather complicated. I believe the reader should first understand the basics beyond cross products and how to calculate them before encountering issues of coordinate systems and vectors versus pseudovectors. Wonder what you think. Oleg Alexandrov (talk) 01:39, 6 May 2007 (UTC)

Alexandrov: Renaming and moving was good, thanks. I liked the previous change you made with the image at the top. I see nothing to change further. The references are also coherent now. For example, if the reader goes to pseudovector, she'll read: "A common way of constructing a pseudovector p is by taking the cross product of two vectors a and b." Thank you for further motivating the simplification; I did not have enough time to make it as clear when I first revised the article and I came back to it today. As I was editing, I saw your posting and then I had to merge three changes... but it was good to see how solid wikipedia is for managing concurrent changes. Edgerck 01:54, 6 May 2007 (UTC)
 * Sorry about editing the article and causing edit conflicts. I was impatient. I am glad it did not make you mad (it can, sometimes :) I am glad we arrived at an agreement. I hope we'll intersect again in the future. By the way, my first name is Oleg, but calling me by last name is fine too. All the best, Oleg Alexandrov (talk) 03:34, 6 May 2007 (UTC)

Oleg: Thanks. Of course, our agreement is no guarantee that every reader-editor of wikipedia will agree. So, I am glad I know you're watching it.

When I first saw the article, for fun, I saw that it defined the cross product to be a vector, even though the existing pseudovector entry at wikipedia correctly said otherwise. It also confused notation with the wedge operator, and made that "Fortunately,..." misstatement. The handedness issue was presented almost as a quirk, rather than an essential property.

On another note, Gibbs is often made to be a villain in math literature for promoting such a cross-creature. However, the usefulness of the cross product -- when correctly used -- is that it makes life simpler for simple things. The flip side is that, up to a certain level, students are not motivated to grasp finer things that might be useful even if just for intellectual growth. Vector analysis (per Gibbs) is presented too often as the "end-all, be-all" of directed quantities. Edgerck 18:38, 6 May 2007 (UTC)
 * Cool. Well, today I took a look at the article, and I rewrote the ==Definition== section again. I wanted the reader to first understand carefully the issue of handedness, before mentioning pseudovectors. So I started with saying that the cross product of two vectors is a vector, that vector depends on the handedness, and then I wrote that it is actually a pseudovector. I hope you don't disagree too much. Wonder what you think. Thanks. Oleg Alexandrov (talk) 21:37, 6 May 2007 (UTC)

Oleg: The top (short) definition is correct. The ==Definition== section misleads the reader now, but I think this can be corrected while still keeping it right. Please let me know when you are done editing. You are like a moving target... Thanks! Edgerck 21:42, 6 May 2007 (UTC)
 * I won't touch that article now. Sorry about being a moving target. I see your point, a pseudovector is not a vector, but you see, if people who are trying to learn what the cross product is are told first thing that the cross product is a pseudovector, they would be confused. Anyway, I'll wait and see how you correct the def. Oleg Alexandrov (talk) 21:52, 6 May 2007 (UTC)

Oleg: Done editing. Students learn the cross product in a RHCS, so making this assumption explicit in the definition makes the definition correct for a vector and allows the discussion to be simpler. Wonder how it reads for you. Edgerck 00:00, 7 May 2007 (UTC)

Changed to use n^ for the unit vector (see minor edits).Edgerck 00:06, 7 May 2007 (UTC)
 * I am very happy with your rewrite. Stating that the coordinate system is right-handed from the very beginning solved all the problems. I think this is much better than anything we had there earlier. Thanks! Oleg Alexandrov (talk) 01:35, 7 May 2007 (UTC)

Oleg: Thanks too. Edgerck 03:38, 7 May 2007 (UTC)

A few comments
This kind of topic is known by experienced editors to be challenging. It will be viewed by many readers/editors, some with minimal experience, some who think they know more than they do, and some true experts. Incorporating expert knowledge as appropriate without confusing naive readers is hard writing, all by itself. Then comes the challenge of fending off ill-conceived but well-intentioned edits. I only take on such challenges when I feel either exceptionally public-spirited or exceptionally masochistic. :-)

The Irish mathematical physicist Hamilton originally introduced the quaternion product, and with it the terms "vector" and "scalar". Given two vectors, u and v (quaternions [0,u] and [0,v]), their quaternion product can be summarized as [−u·v,u×v]. Maxwell used Hamilton's quaternion tools to develop his famous equations, and for this and other reasons quaternions for a time were an essential part of a physics education. But Heaviside on one side of the pond and Gibbs in Connecticut felt that quaternion methods were too cumbersome, often requiring the scalar or vector part of a result to be extracted. Thus the dot product and cross product were introduced — to heated opposition. We now know that quaternions are quite special, and so is the cross product. If we want to work in arbitrary dimensions we would do better to adopt a Clifford algebra, perhaps in the form of a geometric algebra. This is a bit too much to impose on the hapless student trying to make sense of cross products for the first time.

I make the historical remarks partly because they are missing from the article, and partly to establish credibility before making the following objection. It is fundamentally wrong — or at least sloppy — to claim that a cross product produces a pseudovector. That is one way to interpret a cross product. But we may perfectly well define an alternating bilinear function,
 * $$ \operatorname{cross} \colon \R^3 \times \R^3 \to \R^3, \,\!$$

for which no pseudovectors are in evidence. It is when we interpret the cross product, say, as the normal to a surface given by two tangent vectors, that we are in peril. If we want to maintain a certain geometric meaning, we need to understand dependence on coordinate basis.

Another way to come at this is through the Clifford algebra Cℓ3(R) with negative signature, meaning v2 = −||v||2. We can identify the even subalgebra (consisting of scalars and bivectors) with quaternions. With Hamilton, we can then embed R3 in the quaternions, and compute the equivalent of a cross product by killing the scalar part of the quaternion product. Alternatively, we can use the R3 which forms the vector part of the Clifford algebra and create an honest bivector, also known as a dyad or pseudovector. One is not "right" and the other "wrong"; these are choices.

Perhaps the issue of interpretation will be familiar from matrices. Many people are in the sloppy habit of thinking a matrix "is" a geometric operation, such as a rotation. But this is just as silly as thinking a real number is a temperature! We can use a real number for many different purposes, and even as a temperature we have different scales. Likewise, we can use a matrix in many different ways (including alias and alibi actions), and interpretation depends on basis (and inner product).

I happen to think introducing pseudovectors is a confusing waste of time, and only briefly postpones the need for real understanding. We might instead claim that a cross product computes a dual vector, a vector in the space of 1-forms on R3. This view is consistent with using a surface normal (a "normal vector"), n, as an abbreviation for a plane equation, since n·v = 0 describes a plane through the origin comprised of vectors v perpendicular to n.

So, I'm not really satisfied with the current state of the article. But since I am feeling neither sufficiently public spirited nor strongly masochistic just at the moment, I will merely state my views on this talk page for now. --KSmrqT 13:14, 22 May 2007 (UTC)


 * At least for now the pseudovectors are introduced in a way which does not confuse people, that was the point of the debate above. :) Oleg Alexandrov (talk) 15:23, 22 May 2007 (UTC)


 * I'm more or less happy with the state of the article at he moment, the amount of weight given to the distinction between vectors and pseudovectors seems about right. It would also be worth mentioning the 1-forms interpretation as well. Indeed there is scope for an article covering the various dualities which occur. This distinction is useful in practical applications - I was one writting a computer graphics program which transformed surfaces defined as polygons with a normal vector, it took a bit of time to figure out why my normals were all messed up.
 * I do like KSmrq's history of the topic, just the sort of thing an encylopedia article should have. --Salix alba (talk) 17:32, 22 May 2007 (UTC)

In reply to KSmrq and as a way of further explanation. I'm quite aware that this area suffered inconsistencies in the past (especially by Gibbs!). However, the indisputable point is this: if you define a vector as Gibbs did and calculate the cross-product of two vectors so defined, the resulting vector will flip sign when you do, for example, a parity transformation. This is not acceptable either in math (ambiguous) or in physics (the world collapses -- just kidding!), as I explained in cross product handedness discussion. To save Gibbs' vector calculus, the concepts of pseudovectors and (true) vectors were invented and everything works fine (the world does not collapse when you see yourself in the mirror) provided that the special rules explained in the cross product handedness discussion are assured for every equation you use. Further, note that with Gibbs vectors one needs a metric space too, and pesky coordinates are required.

The same problem happens with scalars. If a "scalar" flips sign under under a parity inversion then it is a pseudoscalar, not a (true) scalar. This difference is also important in physics.

BTW, I decided to use the pseudovector (instead of axial vector, or other) terminology because it seemed more extensively used in WP and is usually taught. I thought it was great not to have to talk about "dual vector space" and 1-forms -- that might make the whole subject harder for a beginner. And, in the way it was done, the cross product could be correctly defined, on first sight, as an ordinary (true) Gibbs vector.

Of course, everything works better with Clifford algebra! Happy will be the day when undergrads can learn physics directly using Clifford Algebra. Hope this is useful. Edgerck 18:00, 22 May 2007 (UTC)

Magnitude
The article says that the magnitude of a cross b is given


 * $$\mathbf{a} \times \mathbf{b} = a b \sin \theta \ \mathbf{\hat{n}}$$

Isn't this techincally still a vector rather than a scalar quantity, because it includes $$ \mathbf{\hat{n}}$$ which gives directional information, if only in a generic sense?

EikwaR 22:07, 20 May 2007 (UTC)

Right. The equation defines the cross product (a vector, not its magnitude). However, somebody already corrected the mistake in the article, I guess. Paolo.dL 17:46, 8 June 2007 (UTC)

Notation
Whats with the none-standard notation? —Preceding unsigned comment added by 134.88.60.119 (talk) 18:03, 6 February 2009 (UTC)

Conventional vector algebra
Several authors refer to the algebra endowed with dot and vector product as "conventional vector algebra" (CVA), as opposed, for instance, to geometric (vector) algebra. Moreover, many university textbooks and university course syllabi use simply the expression "vector algebra" to refer to this kind of elementary algebra.

But there is no article on Wikipedia that specifies the name or names of an algebra endowed with a (dot and) a cross product:
 * Simply VA?
 * Cross product algebra?
 * Gibbs algebra?
 * Basic, or elementary, or conventional, or classic VA?
 * Basic, or elementary, or conventional, or classic R3 VA?

Suggestion. Since this kind of algebra is used in basic physics and computer science, and known all over the world by millions of people, wouldn't it be advisable to open a page or insert a section somewhere (here or into the article about cross product) explaining this? As far as I know, much less people know about the differential functions described in this article.

Terminological question. Are there any mathematicians who could give us their opinion about the "strength of association", based on common usage, between these words and the above mentioned vector algebra?
 * 1) Conventional
 * 2) Gibbs
 * 3) Classic
 * 4) Basic
 * 5) Elementary

Also, some of these words may be inappropriate, from an historical point of view (see next section). Paolo.dL 09:19, 25 July 2007 (UTC)

changes needed
"of the coordinate system is not fixed a priori, the result is not a (true) vector but a"

This sentence is not written well, it creates a brick wall at "a priori" and should be edited. —Preceding unsigned comment added by 65.193.87.52 (talk • contribs)

Did Joseph Louis Lagrange know the cross product before it was invented? (Part 1)
Is there anybody who knows history well enough to complete the history section which I just added into the article? (I copied it from a comment by KSmrq). I am extremely curious to know who first defined this kind of vector multiplication. This knowledge, for instance, would allow us to answer the following two questions:

Question 1. Is the above mentioned "conventional vector algebra" (CVA) truly "classic", from an historical point of view? Was the cross product defined before or after Hermann Grassmann defined a similar vector product (called outer or wedge or exterior product)?

Question 2. Was the cross product (also known as Gibbs vector product) defined by Josiah Willard Gibbs (I doubt it) or by someone else much before he was born? As far as I understand from the enlightening comment posted above by KSmrq, Gibbs only promoted the cross product and possibly gave it that name. That's probably the reason why the cross product is also (sporadically) referred to as Gibbs vector product. However, an identical multiplication was part of the quaternion product previously defined by William Rowan Hamilton. Moreover, surprisingly the article on Lagrange's formula seems to imply (if that Lagrange is Joseph Louis Lagrange) that the cross product was known (possibly with a different name and symbol) much before Grassmann, Hamilton and Gibbs were born:

Paolo.dL 09:19, 25 July 2007 (UTC)


 * Grassmann and Hamilton independently came up with their products around 1843, long after Lagrange died. The definition and name of the cross product as an separate entity came from Gibbs, around 1881; Heaviside did not use that name but split out the same product (from quaternions) at about the same time.
 * I don't know the story behind the naming of the triple cross product identity as relating to Lagrange. Keep in mind that many facts were known in component form before there was any "algebraic" form. For example, determinants were studied long before matrices were invented; and Cayley showed the relationship between quaternions and matrices (as we read it today) some years before he defined matrices as such. Euler knew how to describe rotations in 3D, but he certainly didn't use quaternions or matrices or cross products to do it; yet quaternions are sometimes called "Euler parameters" (not to be confused with "Euler angles"!), and it is common to see Euler angles explained using cross products and matrices. --KSmrqT 22:09, 25 July 2007 (UTC)

Before-Hamilton. Thanks a lot to KSmrq for his contribution. He also brilliantly edited the History section. But there's still work to do. Some interesting information is probably missing about the before-Hamilton history. We need information about the genesis of the cross product in his "component form" (see KSmrq's comment above): a set of operations composed of three subtractions of products.

Unsolved paradox. If the author of Lagrange's formula is the famous Joseph Louis Lagrange, he defined the triple cross product two generations earlier than the simple cross product was defined by Gibbs and one generation before Hamilton used it as a component of his quaternion product! Is there anybody who can solve this appearent paradox?

Unanswered questions.
 * Was there some other Lagrange in the history of mathematics (I doubt it)? Or were both Hamilton and Gibbs just using a "product" or set of operations which was already known and studied by Joseph Louis Lagrange, one generation earlier?


 * If Joseph Louis Lagrange is the author of Lagrange's formula, then who invented the set of operations that Gibbs called "cross product"? And how and when and why?


 * In the history section, this is not very clear: the name "cross product" was originally introduced by Gibbs himself or by someone else, inspired by Gibbs's notation?

Paolo.dL 09:16, 26 July 2007 (UTC)


 * Once more, I do not know how Lagrange's name got attached to that identity, but I have stated unequivocally that he died before Hamilton invented the quaternion product, the progenitor of the cross product. The invention of the quaternion product owes nothing to Lagrange. It is one of the most dramatic and well-documented inventions in the history of mathematics, as one can easily verify. There was absolutely no prior notion of "cross product" to draw on, and in any case we know that Hamilton did not do so, because he documents his thinking and how his invention took place. Gibbs and Heaviside both "invented" the cross product. Hamilton wrote Vab for the vector part of the quaternion product of vectors a and b; we would recognize this as the cross product, but Hamilton saw no need to define this composition of operations as a thing in itself. The product of two quaternions is a quaternion; the product of two vectors, under Gibbs, could be either a vector or a scalar. Please read the history for yourself; Tait ridiculed Gibbs' pair of products as a "hermaphrodite monster". I would tell you whether Gibbs himself used the words "cross product", or if that was introduced by his student; but my sources are not clear. If the words appeared in the 1881 lecture notes, or in Gibbs correspondence, that would settle the question definitively; I do not have those at hand. Feel free to do the research and report back. :-)
 * This history is well-trod turf; references to it abound. I am trying to be careful to rely on the most trustworthy sources. For example, a citation of Wilson's Vector Analysis says it was published in 1902, but a photographic reproduction on the web (linked in the article for your viewing pleasure) shows the date as 1901. Gibbs often gets the credit for inventing the name, but I'm just trying to be a little more careful until I see the evidence, or a clear citation for it. But that's a minor point. Maybe it was something Gibbs said in his class lectures but never wrote down, maybe it was a term he used in correspondence, maybe Wilson thought it seemed like a natural phrase. We do know that Gibbs explicitly used the phrase "skew product", and we do know that Gibbs invented this product as a thing in itself.
 * Really, you can read this for yourself! Here is the way it appears in Wilson:
 * The skew product is denoted by a cross as the direct product was by a dot. It is written
 * C = A × B
 * and read A cross B. For this reason it is often called the cross product. More frequently, however, it is called the vector product, owing to the fact that it is a vector quantity and in contrast with the direct or scalar product whose value is scalar.
 * The natural inference is that Gibbs himself used the term "cross product", else why would Wilson say "often called". But I would rather be vague than sloppy.
 * It's fine to ask the questions. Now go dig up sources to answer them. --KSmrqT 20:52, 26 July 2007 (UTC)

NOTE: The discussion about history continues below. The discussion about "questions vs answers" continues in the next section. Paolo.dL 14:52, 1 August 2007 (UTC)

Asking questions vs giving answers
NOTE: This section discusses mainly the method used in the previous section. Paolo.dL 23:32, 30 July 2007 (UTC)

Unfinished job. Thank you. If I could easily find books about the history of mathematics, of course I would dig up and answer. I am not a mathematician. I give answers on articles in my field. However, a well posed and relevant question is as precious as a correct answer. I started a job as well as I could, you offered an outstanding contribution, somebody else will possibly finish our job. That's typical on Wikipedia. Is there anybody who is willing to finish the job we started? Paolo.dL 09:31, 27 July 2007 (UTC)


 * I did not say finding sources was easy. However, you will notice that I have provided online sources, which you or anyone else can discover and read. If you were connected with Yale University, where Gibbs taught, you would have access to material not available to most editors. If you were in Dublin, you could turn up physical copies of Hamilton's work. That would be great, and perhaps make the detective work easier. But a great deal is possible without such special access, as I have shown.
 * For example, you could easily have learned for yourself how quaternions were invented, rather than ask. Or you could try to learn about how the name of Lagrange became associated with the triple cross product identity, and report back on what you were able to discover.
 * Wikipedia has no staff to get things done; we, the editors, must do everything. Questions and problems we have in abundance; what we seek are answers and solutions, and champions to produce them. You started down the right path when you undertook to begin a History section. To be a true champion, you must now do the hard work to complete the task — if you feel that your questions are worth answering. --KSmrqT 17:17, 27 July 2007 (UTC)

Free to contribute. As I already tried to explain on your user talk page about 9 hours ago, it is not up to you to decide what I must do. Your use of the imperative mood (at the end of your 26 July comment) and of the modal verb "must" (in your latest comment) is unpolite and shows that you mistakenly assume to be entitled to impose rather than suggest. If you like, please answer on your user talk page, that I included in my watchlist. This kind of discussion does not belong here. Paolo.dL 18:50, 27 July 2007 (UTC)

Well started job. Notice that without my job the history section would not exist in this article. You wrote it, but without my contribution, you wouldn't have given yours and the history section wouldn't be there. You answered some of my questions because I aroused your curiosity or interest and you felt they were worth answering. Some other mathematician may be able to solve the last enigma, if she/he feels it's worth attention (see below). Otherwise, it will remain unsolved. Paolo.dL 10:07, 28 July 2007 (UTC)

Wilson knew Lagrange's formula! I just discovered that the entire book by Wilson (Gibbs's student) is published on line here in PDF format. I thought it was only a reproduction of the cover and index. Thanks to KSmrq for sharing the link. As you can read at page 74, Wilson (and most likely Gibbs as well) knew "Lagrange's formula", but did not attribute it to Lagrange. A search in the downloaded file showed that the name of Lagrange is never used in that book. In the preface, Wilson gave credit only to Gibbs, Hamilton, Föpps and Heaviside, and wrote that he only rearranged their material: "The material thus obtained has been arranged in the way which seems best suited to easy mastery of the subject." Is there anybody who can solve the enigma and explain the reason why the triple product expansion is called Lagrange's formula (at least on Wikipedia and PlanetMath)? Paolo.dL 20:55, 27 July 2007 (UTC)


 * The peculiar nature of your responses suggests we are having communication difficulties.
 * My use of "must" was conditional, not imperative. I never stated nor implied, in any common understanding of the English language, that you are compelled to answer your questions; I only said that Wikipedia needs champions to get the hard work done.
 * I could hardly have been more explicit in giving you credit when I said "You started down the right path…"; the fact that my words lay dormant on the talk page for so long before they were added to the article underlines my point that our great need is for those who are willing to do the work of researching and editing.
 * Perhaps you do not have a DjVu plugin installed with your browser, and clearly the standard practice of the archive is unfamiliar to you; but on the left side of the item page is a listing of four different formats (DjVu, PDF, TXT, Flip book) and an FTP page, as well as a link to "Help reading texts". The DjVu version is 1&frasl;4 the file size of the PDF, and is the preferred form. You must have a reader, either standalone or plugin, to view this format, just as you must have a reader (or Mac OS X) to view a PDF.
 * For the benefit of mathematics editors, we have compiled a list of mathematics reference resources. The amount of source material freely available on the web today is remarkable; please take advantage of it. (My apologies; I should have thought to mention this earlier.) Those who have never done so may wish to discover for themselves the joy of reading original sources. Yes, it can be difficult to find them, and challenging to interpret old language in modern terms; but I often find it illuminating and humbling and inspiring to see the ideas presented in the words of the masters.
 * Thanks to Paolo for initial efforts to track down the Lagrange connection. A word of caution: PlanetMath and especially MathWorld are not always reliable, as we have found to our chagrin. Use them as places to start, not as the last word on a topic. Moreover, in citing any secondary source, it is dangerous to repeat claims without viewing the original source. For example, suppose Paolo writes that a certain fact may be found on page 74 of Wilson; scholarly practice allows me to repeat that claim attributing it to Paolo, but not otherwise unless I have verified it for myself in Wilson. Why? Because Paolo may have had a bias, or copied the claim from elsewhere, or misread it, or used a different edition, or stated it correctly in his notes and copied it wrong, or had an error introduced at the printer, and so on. A startling number of oft-repeated "facts" turn out to be unsubstantiated. So be careful. But have fun. :-) --KSmrqT 17:09, 28 July 2007 (UTC)

Relay race example. You wrote that Wikipedia needs campions who give answers. I immodestly believe that I am a champion and that I have already won the first lap of a relay race by asking interesting questions... :-) And I believe you are the champion who skillfully grabbed my relay and won the second lap. So, I don't need to run the final lap to show I am a champion. I hope we will win the race as a team. The sprinters who run 4 x 100 relay races are trained for running 100 m, and their powerful muscles are not resistant enough to win a 400 m race. But they are champions.

User feedback. Ideally, people should do what KSmrq suggests. However, readers who ask interesting questions, or reveal relevant weaknesses (e.g. inconsistencies) in articles are precious and should by no means be discouraged. Well posed questions are of great help, they are extremely desirable, they are much better than no feddback whatsoever. In some cases a reader has just two options: no feedback or just asking questions. This happened to me several times while I was reading math articles, because I am not a mathematician. Those who read my questions are free to ignore them, but some may find them very useful as a guide to make their job better. "User feedback" is appreciated in many circumstances, for instance in computer programming. When I write or teach, I love receiving feedback from the readers or students. But only a few students in my class make interesting questions. Similarly, readers who ask interesting questions or reveal relevant weaknesses in articles are rare, but they make the difference and they are welcome in Wikipedia.

Team work. I agree that generally, if possible, answers should be provided rather than just questions. And actually I do contribute with answers elsewhere. However, in this specific case I couldn't, and I don't think I should have. I am not a mathematician. If some mathematician will read my questions and will happen to know the answer (as you partially did), that will save me a lot of time, which I will be able to spend editing elsewhere, and answering questions which I can easily and authoritatively answer. This is a much more efficient way to work than that you suggested. A non-mathematician trying to study mathematics from original sources, in a community containing many expert mathematicians, is a "waste of time and talent". Wikipedia is a community of people working together and helping each other. Very often somebody begins an article as a stub, and many others complete and refine it. Imagine you lead a team of researchers, one of which is mathematician, and you ask to another, who is not a mathematician, to solve a complex equation that the mathematician can solve in a few seconds. That's not an efficient way to manage team work. The "relay race method" worked very well in this particular case: you answered some of my questions because I aroused your curiosity or interest and you felt they were worth answering. Some other mathematician may be able to easily solve the last enigma, if she/he feels it's worth attention. Otherwise, it will remain unsolved. Paolo.dL 10:45, 30 July 2007 (UTC)

Did Joseph Louis Lagrange know the cross product before it was invented? (Part 2)
Wilson knew Lagrange's formula! I just discovered that the entire book published in 1901 by Wilson (Gibbs's student) is available on line here. As you can read at page 74, Wilson (and most likely Gibbs as well) knew "Lagrange's formula", but did not attribute it to Lagrange. A search in the downloaded file showed that Lagrange's name is never used in that book. In the preface, Wilson gave credit only to Gibbs, Hamilton, Föpps and Heaviside, and wrote that he only rearranged their material: "The material thus obtained has been arranged in the way which seems best suited to easy mastery of the subject." Paolo.dL 20:55, 27 July 2007 (UTC)

The enigma and its possible solutions
Joseph Louis Lagrange died much before the cross product was invented (see chronological table above). What is the reason why the triple and/or quadruple product expansion is called Lagrange's formula, or Lagrange's identity? There are two possible solutions for this fascinating enigma: Paolo.dL 10:54, 4 August 2007 (UTC)
 * My hypothesis is that the terminology is wrong and should be abandoned. But this terminology is used in several references (see list above). It might have been mistakenly called that way by some author, and others followed suit without checking original sources.
 * The author of the formula is another Lagrange (Joseph Louis's nepew?), living in the 2nd half of the 19th century. But his work was not credited by Wilson in 1901 (see my 27 July posting).

Call for help (it's easy)

 * If you have access to the database of a math Library, would you mind to check whether a mathematician called Lagrange published papers or books in the 19th or 20th century (most likely after 1880)?
 * Would you mind to consult your own math textbooks (or those in the library of your university) and update the two lists of references that I included above? It would be nice to discover, based on the contribution of many users, what is the oldest book attributing to some Lagrange the triple (or quadruple) product expansion formula. More importantly. please check whether the first name of that Lagrange is given.

Paolo.dL 15:11, 31 August 2007 (UTC)

My edits
Since nobody could justify the name "Lagrange's formula" with an appropriate reference, I substituted it with "Triple product expansion" (used by Wilson in his book, which is most likely the first book containing that formula).

I also moved to Lagrange's identity the sentence about Lagrange's (allegedly second) identity. Notice that only the vector triple product is discussed here. The scalar triple product is not discussed at all (there's just a reference to Scalar triple product in the first sentence of the introduction). I believe there's no reason to discuss Lagrange's (allegedly second) identity here.

Notice that I also proposed the article Lagrange's formula for deletion. Paolo.dL (talk) 21:54, 11 January 2008 (UTC)

Tracing the origin of the expression "Lagrange's formula"
The article Lagrange's formula, that I proposed for deletion, was created by just copying a section of Cross product (see edit summary in history page). The name "Lagrange's formula" is used in the Cross product article since 2002, when it was created (see earliest contribution in the history page). I could not understand who created this article; the first edit summary says it was copied from Vector in 2002; however, the history of Vector, which is now a disambiguation page, starts from 2003.

Wikipedia is not reliable enough as a bibliographic source. In the future, we can add a redirect from "Lagrange's formula" to triple product, provided that someone will find a reliable reference proving that the triple product expansion is related to some Lagrange; in my opinion, it would not suffice to know that someone used this expression in the literature; as far as we know, the name might just be misused by a single author or by the creator of this article.

Right now, we don't even know whether this name has ever been used in a university textbook. If we discover that it was used, we can write in Triple product a sentence like "called Lagrange's formula by ..., although the origin of this name is unknown and controversial (Joseph Louis Lagrange lived about one century before the cross product was introduced by Gibbs and Heaviside)". Paolo.dL (talk) 12:21, 12 January 2008 (UTC)


 * The Vector article is older than 2002, but it was moved to vector (geometry), so the edit history is now there. The edit history is incomplete, of course, because the old Wikipedia software didn't keep a permanent record of edits. The Wikipedia nostalgia site has an old copy of the Vector article, with some older edit history. (This shows, incidently, that the term "Lagrange's formula" was already in the article as far back as 8 November 2001.) --Zundark (talk) 16:05, 12 January 2008 (UTC)


 * Page 1679 of the Encyclopedic Dictionary of Mathematics (Second Edition, 1987) calls the equation [a,[b,c]] = (a,c)b &minus; (a,b)c "Lagrange's formula". --Zundark (talk) 17:42, 12 January 2008 (UTC)

Thank you. Very interesting. But I am puzzled. I am not familiar with the notation used in the Dictionary. I know that  is an inner product, but what about [a,b] and (a,b)? In Wikipedia, I could only find the definition of [a,b] as a commutator... Paolo.dL (talk) 18:13, 12 January 2008 (UTC)


 * [a,b] is the cross product, and (a,b) is the dot product. I don't know why they use this notation, but they define it earlier in their article on vectors, so this is definitely what it means. (They do also mention the usual a×b and a·b notation.) --Zundark (talk) 18:52, 12 January 2008 (UTC)


 * Remark. Although it may seem strange to attribute a 19th century formula to an 18th century mathematician, Lagrange's contribution is not to be taken lightly. Whereas indeed anything explicitly involving a cross product clearly cannot have been due to Lagrange, it is nevertheless possible that Lagrange derived precisely the same identity in geometric terms.  Indeed, it is quite likely that he did and that it was well-known to mathematicians of the 19th century, for Lagrange was responsible for one of the definitive reference works on the tetrahedron: Solutions analytiques de quelques problèmes sur les pyramides triangulaires (1773).  Unfortunately, Gibbs's book Vector Analysis (1901) contains the relevant identity, but virtually no bibliographic details &mdash; a general deficiency of the book as a whole, indeed of nearly all mathematical books written in those days.  It would be nice if someone were willing and able to track down a copy of Lagrange's work to verify exactly how he treats this identity, and if indeed it is deserving of the label.  Silly rabbit (talk) 20:57, 12 January 2008 (UTC)


 * I went ahead and re-added the reference to "Langrange's formula," since it does appear that some sources use this terminology (notably the Encyclopedic Dictionary of Mathematics). However, I support the prod on the article Lagrange's formula since I doubt this term is sufficiently widely used to deserve a separate article.  Silly rabbit (talk) 21:26, 12 January 2008 (UTC)

That's amazing. Indeed, Lagrange was a great master. When I started this discussion, KSmrq warned me that sometimes formulas are introduced in component form before their "algebraic" form is defined. For instance, determinants were known long before matrices were introduced... (see above, comment dated 22:09, 25 July 2007). However, KSmrq also firmly stressed the originality of Hamilton's contribution, based on which the cross product was defined. More recently, I discovered that, in Lagrange's identity, Lagrange used an "algorithm" very similar to a cross product and an exterior product:


 * $$\sum_{i=1}^{n-1} \sum_{j=i+1}^n (a_i b_j - a_j b_i)^2 $$

If a and b are vectors in R3, this expression defines three pairs of "crossed" multiplications, identical to those used for the cross product (although their order and sign is different; there is also a fourth pair of multiplications, which can be ignored). And this was done at least one generation before Hamilton (quaternion product), Grassmann (exterior product) and Gibbs/Heaviside (cross product).

Silly Rabbit gave us a good reason to believe that Lagrange may have written a formula similar to the triple product expansion, in "component form". However, we are not yet 100% sure that he did. If he did, I would love to know exactly how the original identity was written. Paolo.dL (talk) 02:34, 13 January 2008 (UTC)

About tetrahedra. Silly Rabbit aroused my curiosity by writing that Lagrange's study on tetrahedra might be related to the triple product expansion. Thus, I studied the article Tetrahedron, and discovered that the volume of the tetrahedron can be computed using a scalar triple product. However, the triple product expansion ("Lagrange's formula") expands a vector triple product. Thus, I still don't understand the reason why Lagrange may have used and expanded vector triple products. Paolo.dL (talk) 14:19, 13 January 2008 (UTC)


 * Geometrically, the vector triple product formula relates the angle one leg of the tetrahedron makes with the base, and the projection of the leg onto the base. Silly rabbit (talk) 14:27, 13 January 2008 (UTC)

Interesting. Thank you for your enlightening contributions to this discussion. I redirected Lagrange's formula to Triple product. Paolo.dL (talk) 20:50, 13 January 2008 (UTC)

Salix alba and I found three references in which the expression "Lagrange's formula" is used with 3 other meanings. A disambiguation page is needed. See Talk:Lagrange's formula. However, this section is not closed. As Silly Rabbit pointed out, we still don't know exactly if Lagrange's work justifies the attribution to him of the triple product expansion. Paolo.dL (talk) 17:24, 15 January 2008 (UTC)


 * I dug up two of Lagrange's papers in which he uses the cross product quite explicitly. Of course, everything is written in components, but the fact that it is a cross product is unmistakable.  He does give a version of the vector triple product expansion, although in slightly less generality than is given here.  It is possible that he does the full version elsewhere.  I will provide full references shortly.  Silly rabbit (talk) 23:23, 16 January 2008 (UTC)

Great job. Thank you very much, Silly rabbit, for solving this enigma. What you have found is amazing, and totally unexpected (at least for me). I added a section for displaying the "Notes" (otherwise your reference "1" would not appear in the article). However, I am not sure that this is the preferred standard for reference citation in Math articles. Paolo.dL (talk) 11:03, 17 January 2008 (UTC)

Actually, we had a discussion on Talk:Integral about a similar case: the attribution to Newton and Leibniz of the first theorem of calculus. We know, for instance, that a special case of the theorem was published before by someone else. History is gradual. The first theorem of calculus, the theory of relativity, the quaternions and the cross product would have been discovered even without Leibniz, Newton, Einstein, Hamilton, Heaviside and Gibbs. They were "just" the last and most skilled members of powerful relay race teams. I am not sure whether Hamilton knew Lagrange's work on tetrahedra (KSmrq seemed skeptical about this; he explained that Hamilton's discovery is known to be an absolutely original insight), but this makes history more human. Our masters were not extraterrestrial. Paolo.dL (talk) 11:21, 17 January 2008 (UTC)

Regarding the initial definition
Where the article says, "The cross product is defined by the formula a cross b equals a*b*sin(theta)*n...," is it necessary to then say that "theta is the smaller angle between a and b?" I ask because sin(180-theta)=sin(theta) on the given domain (0 <= theta <= 180 degrees). So it would appear (to me, anyway) that we don't need to specify that theta must be the smaller of the angles between a and b. But maybe I missed something obvious? Thanks! CinchBug (talk) 22:32, 8 August 2008 (UTC)


 * The condition is redundant with the restriction 0&le;&theta;&le;&pi;, but I think it doesn't hurt to have an extra layer of redundancy for clarity here.  siℓℓy rabbit  (  talk  ) 22:43, 8 August 2008 (UTC)


 * Is this actually correct? Wouldn't the smaller of two angles always be in [0, 9] ?  B. Mearns * , KSC 01:42, 9 April 2011 (UTC)

In the intro it says "Like the dot product, it depends on the metric of Euclidean space." but then it is not clear how to calculate the cross product in an arbitrary metric space. Can somebody add that formula? Thanks, --ac —Preceding unsigned comment added by 128.115.27.11 (talk) 00:20, 21 October 2009 (UTC)

Math equations display problem
my browser is having difficulty displaying this page. How do I fix this? --Charybdis3 (talk) 01:11, 12 January 2009 (UTC)

Linear algebra or vector calculus?
See Template talk:Linear algebra. I am questioning whether the cross product should be included in the topics related to linear algebra. I have taken an applied college linear algebra course, and the cross product was not covered. It seems that the cross product is more used in vector calculus than in linear algebra. Linear algebra is mostly about arbitrary vector spaces, which includes spaces other than R3. ANDROS1337  19:54, 13 January 2009 (UTC)


 * Without much of a formal math background beyond a standard engineering degree, I tend to agree with you. The cross products seems to be mainly useful for 3-vectors that represent physical quantities, and is mainly a calculus tool. MarcusMaximus (talk) 02:53, 14 January 2009 (UTC)

My Brain Hurts
Look... I know this is a mathematical concept and we try to explain things as simply as possible in an encyclopedia, but the explanations here are so mathy and long that it just makes my brain hurt trying to read it. I can barely comprehend it at all. I still have no idea what purpose or point the cross product serves, nor what use it is. There should at least by some kind of article summary for the layman to understand. I'm not even sure I understand it - does the x actually mean multiplication in this case, because if it did, then the cross-product, in theory, would be impossible. As multiplying a 1x3 matrix by a 1x3 matrix is impossible (Vectors are essentially matrices), you'd think that there would need to be some sort of explanation for this. Otherwise, it just baffles the mind. -- Xander T. (talk) 04:58, 8 March 2009 (UTC)
 * Good question. The $$\times$$ means multiplication in the general sense in as much as the "cross product" is a product. This is different from the outer product of two vectors or the inner product of two vectors ($$\mathbf{u}\mathbf{v}^T$$ and $$\mathbf{u}^T \mathbf{v}$$ respectively assuming column vectors and using matrix notation) or the elementwise product of two vectors. In general, there are lots of things mathematicians call multiplication. The definition is given on this page.
 * That said, the cross product is screwy compared to all other products. This screwyness stems from the fact that the cross product only really makes sense in three dimensions. (For this reason, there are extensions to the cross product, also mentioned on this page.)
 * As for use, the cross product is used for two related things: First, it is a way to find a vector perpendicular to two other vectors. This is used a lot in computer graphics, for example, where you have two vectors on a surface and want a surface normal. The second use is to measure "perpendicularness". The dot product of two unit vectors gives a scalar in the range [−1, 1] indicating how parallel the vectors are, with 1 meaning they are parallel, −1 meaning they are parallel in opposite directions, and 0 meaning they are perpendicular. Likewise the magnitude of the cross product of two unit vectors, $$|\hat \mathbf{u}\times \hat \mathbf{v}|$$ is 1 when the vectors are perpendicular and 0 when they are parallel. If the vectors are not unit vectors, then the resulting vector is also scaled by the product of the lengths.
 * This is why the cross product is used to compute torque. You have a force and a lever arm. If the force is along the direction of the lever arm, then there is no torque; this corresponds to $$\mathbf{F}\times \mathbf{r}=\mathbf{0}$$. If the force and lever arm are perpendicular, then the magintude of the torque is simply $$|\mathbf{F}|\,|\mathbf{r}|$$.
 * Is that helpful? If so we could incorporate some of this into the article. —Ben FrantzDale (talk) 22:02, 10 March 2009 (UTC)
 * That does help a little, but I still find that it would be difficult for the layman to understand. However, having not seen the mention of the fact that it's used to help in torque calculations, that would be an important note to add to the article. I suppose if you think of it in such a way that it's like the dot product of two perpendicular vectors - in order for the dot product to equal 1 for both vectors, it has to stick out in the third dimension. (Holy crap, I think that's our layman's explanation right there! Well, assuming they know the dot product lol.) -- Xander T. (talk) 08:08, 29 March 2009 (UTC)

I think you should add
$$c_i = \varepsilon_{ijk} a_j b_k$$ --> $$c_i = \varepsilon_{ijk} a_j b_k = a_j b_k - a_k b_j$$

Commutative
"i would've made the change myself if i was totally literate about the definitions of algebra, but isn't the vector product conmmutative??" —Preceding unsigned comment added by 190.30.32.14 (talk • contribs) 6 June 2009


 * No, it's anti-commutative, as already stated in the article. --Zundark (talk) 21:50, 6 June 2009 (UTC)


 * If you stick your finger and thumb and second finger out at 90 degrees to each other to represent three vectors, then the cross product of your finger and thumb is along your second finger. If you take the cross product of your thumb and finger, then it's the same as turning your hand around so that you swap over your thumb and finger over, so the result is in precisely the opposite direction.- (User) Wolfkeeper (Talk) 21:57, 6 June 2009 (UTC)

Cross Product and Exterior Calculus
The correct equivalent of the cross product in exterior calculus is

$$ \vec{a} \times \vec{b} = \left[ \star \left( \vec{a}^{\flat} \wedge \vec{a}^{\flat} \right) \right]^{\sharp} $$

Note that the Hodge star is a mapping from differential forms to differential forms. Also, the wedge product is an operation defined only differential forms. See also http://en.wikipedia.org/wiki/Musical_isomorphism.

128.100.5.121 (talk) 16:57, 30 September 2009 (UTC)


 * The "musical" notation is not standard, and the cross-product expression is probably incorrect at musical notation. — Arthur Rubin  (talk) 20:02, 30 September 2009 (UTC)

The cross product in seven dimensions
It says in the Encyclopaedia Britannica, that apart from the trivial cases of zero, and one dimension, that the cross product can only be defined in three and seven dimensions. And true enough, set one up in five dimensions, multiply it out, and it will fail to obey the distributive law.

The Britannica article written in the 1970's says that the proof that it can only exist in three and seven dimensions is very complex, and is at the time of writing, not yet completed. Has it been completed yet?

Also, the three dimensional vector cross product fits with Euclidean geometry and is extremely useful as a language for analyzing certain problems. Has anybody ever tried to visualize the seven dimensional cross product geometrically? Can we do that?

In the three dimensional case, we use vectors i, j, and k. In the seven dimensional case we have to use i, j, k, l, m, n, and o. Has anybody tried setting up the associated computational pair equations? It's a tricky task, but it can be done. I think however that it involves some repeats, in that any particular unit vector may be the product of more than one pair of the other unit vectors. When you've got it, check the distributive law. It is an arduous task that will take pages. Chances are that it will fail, due to a simple error on your own part, because of the shear tedious nature of the task. But if you get your applied maths professor to do it for you, as I did, he will prove to your satisfaction that the seven dimensional case does indeed work. I have such a demonstration of the distributive law for a seven dimensional cross product, that spreads over about three or four A4 pages in handwriting. David Tombe (talk) 10:41, 28 December 2009 (UTC)
 * Did you find the article Seven-dimensional cross product that might answer some of your questions.--Salix (talk): 15:02, 28 December 2009 (UTC)--Salix (talk): 15:02, 28 December 2009 (UTC)

Salix, Thanks alot. I didn't realize that such an article existed. I'll have a look at at, and I'll have to now make a link for it in this article. David Tombe (talk) 02:07, 29 December 2009 (UTC)

Assessment comment
Substituted at 20:22, 2 May 2016 (UTC)

Alternative formulation
Wouldn't it be better to use the symbols for vector norm:


 * $$ \|\mathbf{a \times b}\|^2 = \|\mathbf{a}\|^2 \|\mathbf{b}\|^2 \left(1-\cos^2 \theta \right) .$$

Nijdam (talk) 22:02, 25 January 2013 (UTC)

Orthogonal
In the introduction it says that "It results in a vector which is perpendicular to both of the vectors being multiplied". In the next paragraph it states that "If either of the vectors being multiplied is zero or the vectors are parallel then their cross product is zero."

By my understanding, this makes the cross product result in a vector that is orthogonal to both of the vectors being multiplied, as the zero vector cannot be considered perpendicular. Should the introduction be updated to reflect this?

Nerdfencer (talk) 03:18, 18 April 2013 (UTC)


 * Hi! I actually read this and thought you just need a small clarification. Zero vector has an arbitrary direction so it can be considered both perpendicular and parallel to every vector. So, in this case, it can be considered orthogonal to both the vectors being cross multiplied. There is no problem as such in this article. Hope it is clear now. — S yɛd  Шαм iq  Aнм ɛd  Hαsнм i  (тα l к) 04:01, 18 April 2013 (UTC)

I don't know if this is clear
Hi, just reading this part, where it says: "Another identity relates the cross product to the scalar triple product:" Look for it in the middle, copying and pasting in the search box of your navigator. The formula below that, the last vector "a" is multiplying? scalarly? why not putting the dot in that case? the dot is present in the same formula in another dot product, wich creates me a confusion. — Preceding unsigned comment added by Santropedro1 (talk • contribs) 06:33, 9 June 2013 (UTC)
 * The scalar triple product on the left it is a scalar, and hence multiplies "scalarly". You have not pointed out where "same formula in another dot product" occurs, so it is difficult to get to the source of your confusion.  — Quondum 10:36, 9 June 2013 (UTC)

Cross product elephant joke
Does anyone know the origin of the joke popular (within certain circles)? Q. What do you get when you cross an elephant with a grape? A. Elephant grape sine theta. — Preceding unsigned comment added by 108.225.17.141 (talk) 08:34, 2 October 2013 (UTC)

Linearity
The cross product is defined in terms of the normals of the vectors and the sine of the angle between them. It seems to me that the linearity of the cross product isn't immediately evident from this definition, yet linearity is assumed when the expression in terms of an orthonormal basis is derived. Yasmar (talk) 07:29, 15 November 2013 (UTC)

Jacobi Identity
Cross product is anticommutative, doesn't that REQUIRE it obey the Jacobi Identity? What is added to the article by mention of the Jacobi Identity in the lede? I think its a very minor (and obvious) point. Recommend removal as extraneous, not adding to content.216.96.79.164 (talk) 13:51, 9 January 2014 (UTC)
 * I agree that it should be removed from the lede. The reference to anticommutativity and then a Lie algebra in the next sentence is sufficient to make the point. I have boldly made the change. —Quondum 15:46, 10 January 2014 (UTC)

Define possibly to cross product in other dimensions
The cross product should be able to be defined in dimension of 2^n-1, while n is any natural number, why it can only be defined on 3 and 7 dimension? How about 0,1,15,31,63,127......dimension? — Preceding unsigned comment added by 140.113.136.218 (talk) 07:44, 15 April 2014 (UTC)

undefined 21:18, 15 April 2014 (UTC)
 * See: http://math.stackexchange.com/questions/706011/why-is-cross-product-only-defined-in-3-and-7-dimensions
 * "The dim 3 cross product is to the quaternions as the dim 7 cross product is to the octonions. In some sense, the uniqueness of these cross products is equivalent to some uniqueness (but don't quote me on that) of these two division algebras – Ian Coley Mar 10 at 0:03"
 * "Since the only normed division algebras are the quaternions and the octonions, the cross product is formed from the product of the normed division algebra by restricting it to the 0,1,3,7 imaginary dimensions of the algebra. This gives nonzero products in only three and seven dimensions. This gives nonzero products in only three and seven dimensions and not in dimension 0 or 1 because in zero dimensions there is only the zero vector, so the cross product is identically zero. In one dimension all vectors are parallel, so in this case also the product is identically zero. – Sanath Devalapurkar Mar 10 at 0:04"
 * Sincerely, siNkarma86—Expert Sectioneer of Wikipedia


 * You can also look at Seven-dimensional cross product, the other cross product, which is rather more technical so perhaps more useful for answering this question, and includes a number of references on it.-- JohnBlackburne wordsdeeds 21:49, 15 April 2014 (UTC)


 * Looking at it again the proof I find most approachable (it includes a lot of manipulation but relatively little advanced mathematics) and impressive is this one:
 * Multi-dimensional vector product


 * The proof takes only 3 pages, pages 2 to 4, to proves the product only exists in 0, 1, 3 and 7 dimensions.-- JohnBlackburne wordsdeeds 00:19, 16 April 2014 (UTC)


 * Sedenion15 dimension??? — Preceding unsigned comment added by 140.113.136.219 (talk) 04:09, 17 April 2014 (UTC)

undefined 07:05, 17 April 2014 (UTC)
 * See: http://www.physicsforums.com/showthread.php?t=743097
 * "Now, why doesn't it work with n=15 and the imaginary sedenions? Well, sedenions do not form a division ring. Even worse, there are nonzero sedenions whose product is zero."
 * "....[A] famous theorem by Hurwitz shows that the only normed division algebras are R, C, the quaternions and the octonions. See http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(normed_division_algebras)"
 * "In another sense, there is in fact a cross product in every dimension, satisfying a generalization of all 3 stated properties, provided one allows more than 2 factors in the product. I.e. for vectors in n space, it is a product of n-1 vectors. This is discussed on pages 84-85 of Spivak's Calculus on Manifolds. The product is essentially a choice of a vector normal to the hyperplane spanned by the given n-1 arguments, or zero if they are not independent. It has direction chosen to give the usual orientation of n space, when combined with the argument sequence, and length chosen to satisfy the analog of property 3."
 * siNkarma86—Expert Sectioneer of Wikipedia

Too technical?
There's a technical notice at the top of the article but looking at it I can't see anything glaringly wrong with the article. It uses language I think that's appropriate to the level of the topic, with more technical terms given in common language or explained, detailed explanations and not just formulae, images, links for further details on both more elementary and more advanced topics. It gets more technical later but that's appropriate for the topic which is one that's part of high-school mathematics but has many more advanced applications. Any objections to removing the notice?-- JohnBlackburne wordsdeeds 17:04, 7 October 2014 (UTC)


 * Well, obviously I object, since I just made an effort to start the lead with something a little less technical and you reverted it. I tried to start with language that a high school student might understand:


 * In mathematics, points in three-dimensional space can be represented as three-dimensional vectors pointing from the origin to the point, with each of the three elements of the vector giving a distance along the x, y, or z axis. The cross product or vector product of two such vectors is another three-dimensional vector which is perpendicular to both of them.


 * But with that reverted the article starts with


 * In mathematics, the cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by a × b (where a and b are two given vectors). It results in a vector that is perpendicular to both and therefore normal to the plane containing them. 


 * Leading with binary operation, vector, and normal is going to cause some readers to leave the article right away without having learned anything (and no, they're not going to click on links when they're just trying to see if they can get something out of this article.)


 * Can you think of a compromise version of my attempted opening? Loraof (talk) 17:35, 7 October 2014 (UTC)


 * I reverted your addition as that's not how to start an article. An article almost always starts with the topic as the subject of the first sentence: "An xxx is...". The preliminaries such as "what is a vector?" belong in their own articles, linked by wikilinks. So a definition of the cross product is the right way to start. See WP:BEGIN.


 * As for "binary operation" that is the proper term, it's linked, and it also follows it with "two vectors", which helps clarify what binary means, i.e. an operation on two things. Similarly normal follows [perpendicular, also clarifying that, and providing another way of looking at it. All of these are linked in case the terms are unclear or a reader needs further explanation of the concepts.


 * Articles are meant for a wide variety of readers, so it's appropriate to have both the technical term ("binary operation") as well a the less formal definition ("operation on two vectors". We don't need to go further such as include definitions of such terms or things like vectors; that's what wikilinks and other articles are for.-- JohnBlackburne wordsdeeds 18:10, 7 October 2014 (UTC)


 * I agree that my passage should have started right in on cross product, but I couldn't think of a way of writing it without the sentence getting convoluted. Again I request that you come up with a better version. You seem to agree in principle that articles should appeal to readers at a variety of levels, so I don' t understand why you won't help make this one accessible, at least for a sentence or two, to high school students. Telling people on the talk page that readers should click links in order to understand the basic idea at the beginning of an article is not going to affect the fact that if they can't even get through the first sentence without clicking in order to even understand the gist of it, we're going to lose them immediately. Linked terms with no definition here can wait for a couple of sentences. That's how one appeals to readers at various levels -- start easy and then get technical. Loraof (talk) 18:59, 7 October 2014 (UTC)
 * As I noted above I think the current version works well, having reviewed it before starting this section. If you are unhappy with it then it's up to you to suggest improvements, or to say more precisely how it's wrong. But bear in mind the guideline WP:BEGIN for the first sentence (and associated gulidelines for the introduction found on that page), and that it's the whole encyclopaedia that's meant to provide encyclopaedic coverage, not one article. It's not only acceptable but expected that readers who find one article too hard look to other articles for explanation, via links. This is especially true in mathematics, where more advanced topics build upon earlier ones, and there are many articles that are unaccessible to the average/non-mathematician reader or even the high-school educated reader.-- JohnBlackburne wordsdeeds 19:42, 7 October 2014 (UTC)

Operation, or result of the operation?
The first and third paragraphs of the lead, and part of the second paragraph, use the term "cross product" to mean an operation. But pretty much everywhere else in the article uses the term to refer to the vector result of the operation. If both uses are valid, the opening sentence should be adjusted to mention that (maybe just with the added phrase "or its vector result". If there is another preferred term for the operation itself, that should be used where appropriate. Loraof (talk) 21:40, 7 October 2014 (UTC)


 * The cross product is an operation, a binary one that acts on two vectors. Its result is a vector (or in some contexts a pseudovector). So "the cross product a × b is a vector" is correct. As is "their cross product is a vector". Can you point out where it says the cross product refers to the vector result, as I have not noticed any instances of this.-- JohnBlackburne wordsdeeds 22:36, 7 October 2014 (UTC)


 * You say So "the cross product a × b is a vector" is correct. As is "their cross product is a vector". Can you point out where it says the cross product refers to the vector result? In your two sentences preceding your question. And repeatedly subsequently in the article. Loraof (talk) 01:24, 8 October 2014 (UTC)
 * but those are saying the result is a vector. "the cross product a × b" means the result of the calculation. When you write "the calculation xxx is" you usually mean its result. So you'd write "the cross product of a and b is a vector" much as you'd write "the sum of three and two is a number" (i.e. 3 + 2 = 5, a number). "the cross product of a and b" is another way of writing "the cross product a × b", or if you know it's two vectors a and b you could just write "their cross product". Different ways of writing the same thing maybe, but that's part of good writing, avoiding too much repetition.-- JohnBlackburne wordsdeeds 01:47, 8 October 2014 (UTC)
 * I've copyedited the article to be more explicit, and hope I have not made it too cumbersome in the process. It is part of language to essentially elide the implied phrase "the result of taking"; this does not make the term "cross product" refer to the result; it only means that the sentence has been abbreviated. —Quondum 03:57, 8 October 2014 (UTC)
 * I've copyedited the article to be more explicit, and hope I have not made it too cumbersome in the process. It is part of language to essentially elide the implied phrase "the result of taking"; this does not make the term "cross product" refer to the result; it only means that the sentence has been abbreviated. —Quondum 03:57, 8 October 2014 (UTC)


 * This is merely a philosophical discussion. The cross product, if you wish, is just the operation, symbolized by '×'. The cross product of two vectors is again a vector. It is a product, produced by the operation. I hope you will restore the original text on this theme. Nijdam (talk) 20:24, 8 October 2014 (UTC)


 * I don't see that as an improvement at all. It's unnecessary except where it's makes things more verbose or is just wrong. E.g. "If the vectors have the same direction or one has zero length, then their cross product produces zero" ? Their cross product is a vector. How does a vector produce zero? Their are ways to phrase it that make sense but then are unnecessarily verbose. As I wrote at Talk:Dot product this is basic English. One could say "the sum of three and five is eight" or "their sum is eight", or even "their sum is zero". "their cross product is zero" works the same way and clear and unambiguous, unlike "their cross product produces zero" which doesn't make sense.-- JohnBlackburne wordsdeeds 23:49, 8 October 2014 (UTC)


 * I see it as a linguistic distinction between two things: the cross product (an operation) and the cross product of two vectors (the result of the operation applied to the vectors, which is a vector). Thus, the term "cross product" means two different things, usually cued by sentence structure (where even the "of two vectors" can be silent). I agree that my change has made it unnecessarily clumsy. I made a number of edits that were unrelated (format) at the same time, which I'd like to keep. Allow me to clean this up tonight (quite probably by reverting and reapplying selective edits, so that we can easily track changes). —Quondum 00:02, 9 October 2014 (UTC)

Okay, we're essential back to where we started. So let's see whether we can sensibly clear up the interpretation of the language so that a reader does not raise the same objection as Loraof has. Any ideas how we might do that, perhaps in the first two sentences of the lead? —Quondum 04:08, 9 October 2014 (UTC)


 * Thanks Nijdam for this edit; I think it does what I had in mind in a natural way. —Quondum 13:42, 9 October 2014 (UTC)


 * One more way of thinking of it, mostly for the benefit of the original poster. I think we all agree the following is correct (using an example from dot product as it's easier):
 * $$\mathbf A\cdot\mathbf A = \|\mathbf A\|^2.$$
 * But all that says is the dot product of a vector with itself equals the square of its magnitude. And another less formal but correct word for "equals" is "is". Put the other way round, if you would say something is [the same as] something else you'd use "=" to write that down. So "the acceleration due to gravity is nine point eight metres a second squared" becomes
 * You might say "equals" when saying this in words but "is" seems as likely. "=" is being used the same way in this article, so it's also correct to say e.g. "the cross product a × b is a vector".-- JohnBlackburne wordsdeeds 14:30, 9 October 2014 (UTC)
 * You might say "equals" when saying this in words but "is" seems as likely. "=" is being used the same way in this article, so it's also correct to say e.g. "the cross product a × b is a vector".-- JohnBlackburne wordsdeeds 14:30, 9 October 2014 (UTC)

Geometric Meaning
A = $$\left\| \mathbf{a} \times \mathbf{b} \right\| = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \sin \theta. \,\!$$ Why ||a|| ||b|| used? I don't feel the requirement mathematically. 117.198.178.189 (talk) 17:50, 15 May 2015 (UTC)


 * They make the vectors into scalars, i.e. they get the scalar length of the vectors, which is needed for the formula. It does not make sense otherwise.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 18:24, 15 May 2015 (UTC)


 * a is vector. |a| is the magnitude (non-negative) so why ||a|| is required. ||a|| and |a| are the same. 117.198.178.189 (talk) 19:08, 15 May 2015 (UTC)


 * It's just a different way of writing the same thing. See e.g. Norm (mathematics).-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 19:19, 15 May 2015 (UTC)


 * Then why do we use two of them. In that formula, I guess, very huge amount of audience have seen |a|. And it would be better so that everyone feels normal. 117.198.178.189 (talk) 19:24, 15 May 2015 (UTC)


 * It is useful to distinguish between when the norm of a vector and the absolute value of a real number is meant. However, this article was inconsistent in this respect; I've cleaned it up. —Quondum 20:09, 15 May 2015 (UTC)

Representation as a determinant
This deletion has the edit summary (Undid revision [...] no, sorry, that is wrong). It does not seem to be correct to claim that the cross product can be expressed as "an actual determinant" (which would be implied by the deletion), and so I do not understand the reason for the deletion. Determinants are normally only defined for matrices over fields, and may be extended to matrices over commutative rings (as indicated here). In fact, the reminder that the determinant notation is purely a mnemonic to remember the form of the result seems to be exactly appropriate in the article. — Quondum 03:16, 19 May 2013 (UTC)


 * I agree with you. This kind of "determinant" only makes sense if one row is made up of vectors and the other rows are made up of scalars, and I don't know of any other context where such a beast is useful. On the other hand, I don't think the article needs an entire paragraph explaining the objection. Bobmath (talk) 03:58, 19 May 2013 (UTC)


 * Is the qualification "formal" used in the article enough of a caveat? Bobmath (talk) 04:10, 19 May 2013 (UTC)


 * Good observation. Both the determinant and the matrix are "formal" in this sense, so I have adjusted the wording hopefully to reflect this without being cumbersome while qualifying the determinant itself more obviously. Since "formal" is an ambiguous term, I've linked to the intended meaning using the link that you identified.  You also have a point that the additional paragraph may be excessive; inline qualification is probably best; if more is needed, then a footnote would be appropriate. — Quondum 14:44, 19 May 2013 (UTC)


 * The problem with the determinant representation of cross products is due to a simple error. The determinant always gives ixj=k regardless of the orientation of the coordinate system.  It is fundamentally inconsistent with the right hand rule for determining the direction of the cross product.  It stems from replacing ixj by k.  If you do not do this, the determinant is multiplied by i.(jxk). This corrected expression is true regardless of the coordinate system orientation and is consistent with the right hand rule.  This is good in physics since it means that the magnetic field transforms exactly like the electric field under any change in coordinate system!  No more pseudo-vectors! Jcpaks3 (talk) 16:42, 26 September 2015 (UTC)

Anti-commutativity
The cross product is anticommutative (i.e. a × b = −b × a)

Sorry to be pedantic, but anticommutativity is the property a × b = −(b × a). In the case of the cross product these happen to be equivalent, since −b × a = −(b × a) always, but the definition as stated is incorrect. 2.24.119.101 (talk) 01:04, 9 March 2016 (UTC)

Xyzzy
The Mnemonic section claims there are two ways to remember the order of the terms. Although it makes no claim that there are only two ways, nor that the two described are in any way "best", I am wondering whether the way I've always remembered the order is worth including. (If others share my method, then possibly it's not Original Research). It's simple x = yz is in alphabetical order and then yz is reversed (mirror image/reflected) giving zy as the third term. It requires you to remember that 1. order for 1st equation starts alphabetically 2. the 3rd term reverses indices of 2nd and 3. other two equations are forward cyclic permutations of 1st equation. Arguably, this is as compact as remembering xyzzy since both encode x = yz|zy, but doesn't require the recall of whether it is xyyzy, xyzyz etc. Seems to me which is easier depends on whether an individual's mind finds symmetries easier to recall than words. Just a thought.71.29.173.173 (talk) 15:25, 16 July 2016 (UTC)

Non Euclidean Coordinates
I think providing the cross product succinctly in the two common non-Euclidean coordinate systems (Spherical and Cylindrical) might be useful.71.29.173.173 (talk) 15:32, 16 July 2016 (UTC)


 * Á good idea. It may be of interest to go further and give the general curvilinear coordinates form also. Borisenko and Tarapov have it. 'M'&and;Ŝc2ħεИτlk 16:42, 16 July 2016 (UTC)


 * Actually, they state the result in an oblique coordinate system. 'M'&and;Ŝc2ħεИτlk 10:02, 17 July 2016 (UTC)

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Definition
I agree wholeheartedly with the poster who attempted to include the determinant-like definition of the cross product. It has the advantage of directly computing the components of the resultant vector, unlike the definition given, which gives no hint as to how to compute n, the unit vector mutually orthogonal to the two whose cross product is being taken.

Moreover, the determinant-like definition is employed in practice. From my library shelf:

Margenthau & Murphy, The Mathematics of Physics and Chemistry, 2nd Edition, Van Nostrand, 1962, p. 44

A. Taylor, Advanced Calculus, Ginn, 1955, p. 289

C.R. Wylie, Advanced Engineering Mathematics, 3rd Edition, McGraw-Hill, 1966, p. 537

This post was prompted by a Stackexchange post that used the definition from this Wikipedia article, but without bothering to note the presence of the unit vector n. The formula in the article, used in that sense, is commonly used in practical applications only to calculate the magnitude of the resultant. — Preceding unsigned comment added by SCS137 (talk • contribs) 16:11, 4 July 2018 (UTC)
 * which "poster"?--Jasper Deng (talk) 18:39, 4 July 2018 (UTC)

2 severe misleading aspects
The article contains two severely misleading aspects.

First, the section on cross product and handedness talks about bases of mixed vector types, which is a flawed concept and not expressible in algebraic vector space. It is a misleading intuition.

Second, while the idea is correct, that the cross product needs a notion of handedness, the phrase requiring a handedness of a coordinate system is wrong. There is such a notion as an oriented vector space (definable, for example, as a pair consisting of a vector space and a basis, or a volume form, or a determinant, or a similar construct). However, no notions in a vector space should depend on a coordinate system as such. Moreover, in a vector space per se (without an orientation concept), it is not possible to define the notion of a left or right handed coordinate system - this needs fixing an orientation at first. — Preceding unsigned comment added by 217.95.166.244 (talk) 17:51, 1 December 2018 (UTC)


 * I agree with you that earlier in the algebraic vector space there was no concept of mixed vectors. But why is the proposed vector model a fallacy?


 * Let's look at the reasons why you need to change the theory of vectors:


 * 1. The theory of vectors, using only rectilinear vectors, does not describe angular directions at all.


 * 2. Use rectilinear directions (cross product of vectors) to describe the angular direction, leads to a conflict of these directions.


 * 3. To solve this conflict, a “right-hand rule” was invented. As a result, they dissolved the bureaucracy, which complicates the solution of a simple question - which way is the rotation directed?


 * 4. Very often the “right-hand” rule is inconsistent with the chosen coordinate system.


 * 5. If the result of a cross product is a rectilinear vector, then it is necessary to separate it (by description, properties, etc.) from other rectilinear vectors and describe their differences. That is, you need to complement the theory of vectors.


 * 6. The theory of vectors should have a vector division method, because without it, there is no sense at all to use vector theory to solve applied problems. For example, in the strength of materials, formulas of a general form are still used, because they cannot find a force from a torque in coordinate form. As a result, constantly confused with the directions (signs).


 * Your second point is fair. The direction of the angular basis vectors should not be tied to the coordinate system. Probably, the direction of the angular basis vector should depend only on the direction of two basic perpendicular rectilinear vectors. And if in some exotic vector space it is impossible to construct two perpendicular basis vectors, then in this vector space it is necessary to revise the rules for the execution of point, cross vector products. Ujin-X (talk) 10:29, 20 December 2018 (UTC)

Usage of color in equations
Currently, the section Computing the cross product expresses unit vectors using color notation. I think this should be change. First, the color is nowhere referenced in the expository paragraphs surrounding these formulae. Second, the colors conflict with the figure accompanying this section. Third, usage of color may cause accessibility issues, particularly for those who are color blind or with poor vision. Fireballs619 (talk) 00:16, 6 June 2019 (UTC)