Talk:Cross product

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:

• |x×y|² = |x|²|y|²(1-(x·y)²)

This isn't true in 3 dimensions, so I doubt it's true in 7. Note that x·y is not the cosine of the angle between x and y. AxelBoldt 02:06 Apr 30, 2003 (UTC)

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 — 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?

$\mathbf {i} (a_{2}b_{3})+\mathbf {j} (a_{3}b_{1})+\mathbf {k} (a_{1}b_{2})-\mathbf {i} (a_{3}b_{2})-\mathbf {j} (a_{1}b_{3})-\mathbf {k} (a_{2}b_{1})$

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

$\mathbf {i} (a_{2}b_{3}-a_{3}b_{2})-\mathbf {j} (a_{3}b_{1}-a_{1}b_{3})+\mathbf {k} (a_{1}b_{2}-a_{2}b_{1})$

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#Geometric meaning help? Oleg Alexandrov (talk) 02:36, 1 February 2006 (UTC)

Or the formula

a × b = [a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁].

in the section Cross product#Matrix notation. 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#Cross product 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,

$\operatorname {cross} \colon \mathbb {R} ^{3}\times \mathbb {R} ^{3}\to \mathbb {R} ^{3},\,\!$

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. --KSmrq^T 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

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

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:

Joseph Louis Lagrange	1736 - 1813
William Rowan Hamilton		1805 – 1865
Hermann Grassmann		1809 - 1877
Josiah Willard Gibbs			1839 - 1903
Oliver Heaviside			1850 - 1925

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. --KSmrq^T 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. --KSmrq^T 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. --KSmrq^T 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.

1. 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.
2. 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.
3. 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 ¹⁄₄ 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. :-) --KSmrq^T 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 Wikipedia: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)

References

Not attributing to Lagrange the triple (or quadruple) product expansion

• Wilson, Edwin Bidwell (1901), Vector Analysis: A text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs, Yale University Press

Attributing to Lagrange the triple (or quadruple) product expansion

• Lagrange's formula on Wikipedia. In this article, two formulas or identities are attributed to Lagrange:

Lagrange's identity 1: a × (b × c) = b(a · c) − c(a · b)

Lagrange's identity 2: $|a\times b|^{2}+|a\cdot b|^{2}=|a|^{2}|b|^{2}$

• Triple cross procuct on PlanetMath
• W. Kahan (2007). Cross-Products and Rotations in Euclidean 2- and 3-Space. University of California, Berkeley (PDF).. From page 3/25:

"Triple Cross-Product: (p×q)×r = q·p•r – p·q•r" (not attributed to Lagrange!)

"Jacobi’s Identity: p×(q×r) + q×(r×p) = –r×(p×q)"

"Lagrange’s Identity: (t×u)•(v×w) = t•v·u•w – u•v·t•w" (attributed to some Lagrange)

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:

• 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).

Paolo.dL 10:54, 4 August 2007 (UTC)

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#Vector 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 − (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 <a,b> 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 — 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#Vector 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≤θ≤π, 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 <nowki>[0, 9]</nowiki>? 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)

stig

cross prodfgrr is stuped —Preceding unsigned comment added by 71.191.195.17 (talk) 01:27, 17 October 2008 (UTC)

Math equations display problem

my browser is having difficulty displaying this page. How do I fix this? [1] --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)

The seven dimensional cross product

This article needs to be merged with the seven dimensional cross product. The material from the seven dimensional cross product should be brought over to this article, and the latter article deleted. The material from the latter article should be put in a special section entitled 'Seven Dimensional Cross Product'.

Anything to do with the equation |a×b| = |a|×|b|sinθ only relates to the 3D case, and hence should only appear in the sections about the 3D cross product.

Does anybody object if I go ahead and bring the seven dimensional material over to this article? David Tombe (talk) 09:47, 2 January 2010 (UTC)

|a×b| = |a||b|sinθ applies in both three and seven dimensions. It's equivalent to one of the conditions, |a×b|² = |a|²|b|² - (a·b)² from the Pythagorean trigonometric identity. So it's true in both 3 and 7 dimensions.

but as for merging, no that would not be appropriate. This article is on the common vector operation, as per WP:COMMONNAME. When mathematicians and non-mathematicians use the name "cross product" they are almost always referring to this product, not anything else. For most people it's the only one they know. The seven dimensional one is pretty obscure and most people never come across it (I learned of it after finishing my degree). Including it in here would just confuse people, as it's not a mainstream part of vector algebra, which is largely concerned with 3D, and it is also not as well defined as in 3D. --John Blackburne (words ‡ deeds) 10:09, 2 January 2010 (UTC)

I fully agree with JohnB. The seven dimensional obscurity deserves an honorable mention under the generalisations section. It already has the mention. DVdm (talk) 11:03, 2 January 2010 (UTC)

John, Fair enough. But the seven dimensional cross product should at least get a more explicit link. As it stands now, we have to guess that the link to 'seven dimensions' will open out to 'seven dimensional cross product'. Anyway, I note that you say that

|a×b| = |a||b|sinθ applies in both three and seven dimensions.

The way I was taught it was that the equation |a×b| = |a||b|sinθ follows as a consequence of a×b satisfying certain conditions such as the Jacobi identity and the distributive law. The proof involved three dimensional geometry.

If you think that the equation |a×b| = |a||b|sinθ holds in either one or seven dimensions, I'd be fascinated if you could explain to me the meaning of the angle θ in either one or seven dimensions. Could we have a seven dimensional square? I read something in one of these two articles about seven dimensional Euclidean geometry. Now there is a concept I had never heard of before! Can you please explain to me what angle θ means in seven dimensions? David Tombe (talk) 11:40, 2 January 2010 (UTC)

Huh? Don't you know that in n dimensions, the angle θ between two vectors is defined as arccos( (a·b) / |a||b| ), which you can write as arcsin( √( 1 - ( (a·b) / |a||b| )² ) )? Not all that fascinating if you ask me. DVdm (talk) 12:06, 2 January 2010 (UTC)

The angle is defined in seven dimensions as in two and three. You can use e.g. the dot product to calculate it. This can be used to relate the above formula to |a×b|² = |a|²|b|² - (a·b)² – just replace the dot product and use the Pythagorean trigonometric identity.

In one it's a bit less obvious, until you realise that all lines in one dimension are parallel, so θ is zero or π, and so sin θ is always zero. The product is zero always, i.e. trivial.--John Blackburne (words ‡ deeds) 12:03, 2 January 2010 (UTC)

John, As regards your answer about the angle θ in seven dimensions, we'll just have to agree to differ. I cannot remotely imagine what an angle means in seven dimensions. Neither can I imagine Pythagoras's theorem in seven dimensions. David Tombe (talk) 12:07, 2 January 2010 (UTC)

An angle between two vectors in 7 dimensions means the arccos of the dot product of the vectors divided by the norms of the vectors. That is what it means in n dimensions for any n. I'm really amazed you didn't know that. DVdm (talk) 12:18, 2 January 2010 (UTC)

It's just an algebraic theorem which can be applied here, using fairly elementary algebra. I suggest you try it before dismissing it.--John Blackburne (words ‡ deeds) 12:10, 2 January 2010 (UTC)

DVdm, With cosine involved, I think we're back to tautologies again. Everything to do with cosine begins on a 3D premises. I don't think that the physical concept of 3D space can be extrapolated to 7D that easily. Do we have the same π? Try to imagine seven mutually orthogonal axes. How do we fit 360 degrees in there? I know a pure maths professor who was in my form at school. He is a specialist in 'space in n dimensions'. He couldn't answer when I asked him to explain the meaning of angle in 7 dimensions. He also just happens to be the one who taught me the importance of the distributive law in making a cross product useful. Everything to do with the concept of angle originates within the context of our perception of 3D space. I have often believed that when pure mathematicians extrapolate things to their favourite 'n dimensions' that it is an extrapolation beyond reality. It is were maths leaves the real world. David Tombe (talk) 12:30, 2 January 2010 (UTC)

You really continue to amaze me. DVdm (talk) 12:45, 2 January 2010 (UTC)

DVdm, You can have the cosine. But what does the angle itself mean? David Tombe (talk) 12:48, 2 January 2010 (UTC)

I didn't say anything about cosines. I said something about the arccos of a real number. But anyway, tell your pure maths professor that it produces for instance the "real world angle" in the 2-dim subspace spanned by the two vectors. DVdm (talk) 13:48, 2 January 2010 (UTC)

Inner product space#Norms on inner product spaces --John Blackburne (words ‡ deeds) 12:50, 2 January 2010 (UTC)

And WP:RD/MA --John Blackburne (words ‡ deeds) 12:53, 2 January 2010 (UTC)

Yes, I think that educating an interested lay person in elementary aspects of vector spaces is a bit off-topic on this talk page. O.t.o.h referring to the reference desk can only point back to where we have been pointing already. DVdm (talk) 13:48, 2 January 2010 (UTC)

It appears to me that you are both overlooking something very fundamental. The 3D cross product involves 3 components in any operation, and it can be used to describe realities in 3D space. The same analogy does not exist with the 7D cross product. In any given operation, we are still only using 3 components, and the linkage to 3D space has been lost. As regards 7D space, if such a thing actually exists, the analogy with the 3D cross product's relationship to 3D space does not exist.

In the sources, any proof that the 3D cross product obeys the relationship |a×b| = |a||b|sinθ will be explicitly in relation to the 3D cross product's linkage to 3D space. There is no proof in the literature that the 7D cross product has got any corresponding equivalent to the equation |a×b| = |a||b|sinθ in relation to 7D space. It was an over extrapolation on the part of whoever wrote the article, to write it in such a way as to imply that |a×b| = |a||b|sinθ was a general result for cross products in 1,3, or 7 dimensions. This was of course no doubt due to the fact that whoever wrote the article was clearly writing it exclusively with the 3D cross product in mind.

What needs to be done it is to make a clear segregation in the article between the 3D case and the 7D case.

While on the subject, let's imagine a 3D rotation. We'll fix the k axis in space and rotate about the k axis in the ij plane. The concept of angle will be understood in that context. Now let's imagine a 7D situation. Once again, we'll fix k in space. But this time we could define rotations in quite a few planes. And if we take the general cross product a×b where a and b are defined in terms of all seven components, then we will be completely lost. Because once we have established the rotation axis, we will have a headache trying to decide which plane to take our angle from.

A truly 7D cross product which would link to a real 7D space by the same analogy with the 3D case, would have to involve all 7 components in any given operation. We do not have a truly 7D cross product in that sense.

The 7D cross product that we do have, is somewhat of a misnomer. It is 7D only in the sense that it involves 7 components. But these seven components do not actually represent unit vectors in any space. David Tombe (talk) 15:07, 2 January 2010 (UTC)

Not much more I can say. The cross product is defined in 3D and 7D. It can be proven to exist in only those dimensions, subject to the restrictions in the definition. All this is on the 7D cross product page. There's no problem with doing it 7D as vectors, angles, etc. are all defined in 7D, identically to 3D (as an angle is a 2D thing it needs defining in more general dimensions). Apart from the two cross product pages there are the links we've given above and the various wikillinks and references on the pages, where it is explained in much more detail at all sorts of levels of expertise. --John Blackburne (words ‡ deeds) 15:32, 2 January 2010 (UTC)

"The 7D cross product that we do have, is somewhat of a misnomer." ==> This reminds me of "The speed of light is somewhat of a tautology...".

David, I advise you to avoid this kind of alley. Again, your lack of expertise is not the root problem here but it isn't really helpful, given your obvious failing to know how angles are defined in n-dimensional vector spaces yet attempting to reorganise articles about cross products in 7-dim spaces, and at the same using the talk page to complain about "extrapolations beyond reality" and "where maths leaves the real world". You seem to be insisting to push some point resulting from what looks like a shallow synthesis of a few sources. This is the kind of talk page disruption that brought you in trouble before, so perhaps you shoud be careful. DVdm (talk) 16:51, 2 January 2010 (UTC)

Oh I see. This is disruption is it? In that case I better leave you to it. David Tombe (talk) 17:07, 2 January 2010 (UTC)

A source to back up the idea that no source is required

I've got a source here [2] which more or less backs up my viewpoint that no source is necessary to back up the view that the reason why cross product doesn't work in 5D is because it doesn't obey the distributive law.

The source states that students may enquire as to whether or not a cross product exists in dimensions other than 3. The source states,

This note points out that by proving the elementary propositions and theorem below, students may answer this question themselves.

In other words, if the basic axioms such as distributive law, and Jacobi identity hold, then we will only have a result in 1, 3,and 7 dimensions. As I have been repeatedly saying, this fact is too basic to require a source. But although it is basic,it seems to have been overlooked by some of the editors here.

In matters to do with maths, first principle arguments are often sufficient in the absence of sources. David Tombe (talk) 05:47, 4 January 2010 (UTC)

The note suggests the proof is "elementary" for "students" but does not say how elementary. It also does not give the proof, but "indicates" them. Lastly it says nothing about the distributive law, saying instead that the proofs will need "knowledge of" various things, mostly related to orthonormality. As even if you pay $12 you don't get a proof, just an indication of one, it's a lot less useful as a source than the two proofs on Seven-dimensional cross product which are complete and free. --John Blackburne (words ‡ deeds) 09:53, 4 January 2010 (UTC)

"... which more or less backs up my viewpoint that no source is necessary to back up the view that ..." ==> This is probably the best example of a combined case of wp:NPOV, wp:SYN and wp:SOURCE I have ever seen here. That phrase should be taken as an example in various policies and guidelines. DVdm (talk) 10:19, 4 January 2010 (UTC)

John, You were reading the wrong bit. The bit that I was referring to is the quote that I made in italics above, and the sentence immediately following it. I was not talking about any proofs. You are the one who keeps changing the subject to proofs. I was talking about the fact that it is easy to see how only 1D, 3D, and 7D satisfy the basic axioms. One of those basic axioms is the distributive law, and that happens to be the crucial axiom in this issue. But perhaps then you would prefer to put something more vague and general into the main article along the lines of 'the 5D and the 15D cross products don't work because they fail to satisfy the basic axioms'. David Tombe (talk) 11:27, 4 January 2010 (UTC)

The source says nothing about the distributive law--John Blackburne (words ‡ deeds) 11:38, 4 January 2010 (UTC)

David, you might consider stopping point-push-disrupting this talk page. You are behaving here (and on Talk:Seven-dimensional cross product) like you were behaving on Speed of light, which earned you a year long ban on physics related articles. What you are doing could lead to an extension to mathematics related articles, or worse. DVdm (talk) 11:49, 4 January 2010 (UTC)

John, The source talks about axioms. The distributive law is one of the axioms of the 'cross product'. A 5D cross product does not satisfy that axiom. That is why we cannot have a 5D cross product. I am very surprised that you didn't know all this. David Tombe (talk) 11:57, 4 January 2010 (UTC)

Distributivity

The derivation of the formula for the cross product depends on the distributive property, but I'm not aware of a simple proof of distributivity. The article should either include a simple proof, or briefly sketch a hard one, or give a reference. But it should at least acknowledge that there's some work to be done.—Preceding unsigned comment added by 129.89.14.247 (talk • contribs) 20:09, 29 January 2010

It doesn't need to be proved as it's part of the definition: it's required to be a Linear map which implies distributivity. This is stated explicitly in the Seven-dimensional cross product article, which is more concerned with that particular generalisation. That together with the orthogonality and the mangnitude being ab sin θ define the cross product entirely. It's not so important in 3D as for most people it's just the product given by the formula, it could be clearer though in 3D so something about the conditions should probably be included. --JohnBlackburne^words_deeds 20:25, 29 January 2010 (UTC)

The Lagrange Identity in seven dimensions

John, I think you had better explain your reversion. We have just spent a few days arguing about this on the talk page of Seven dimensional cross product. I had been mislead in part because of this article which restricted this particular form,

$|\mathbf {a} \times \mathbf {b} |^{2}+|\mathbf {a} \cdot \mathbf {b} |^{2}=|\mathbf {a} |^{2}|\mathbf {b} |^{2}.$

of the Lagrange identity to 3D. After working it out from first principles I finally conceded that you were correct and that it holds in 7D also. I came here to fix this article in that respect. You have now reversed your position. You had better explain yourself. David Tombe (talk) 10:54, 20 April 2010 (UTC)

There is nothing wrong with either article that needs 'fixing', it is only that you (still, clearly) do not understand it. Lagrange's identity is only equivalent to the above ("Pythagorean") identity in 3D, as its article says. Further this article is about the cross product in 3D, so should not have higher dimensional results inserted at arbitrary points. It links to the 7D version in the lede, has a section on higher dimensions with links further down.--JohnBlackburne^words_deeds 12:01, 20 April 2010 (UTC)

John, Are you sure you understand the subject? On this page, you are saying that it is 3D only. But over at Seven dimensional cross product you are saying that it is both 3D and 7D. Your argument about this being the 3D page doesn't wash because you also did the same reversion on the more general Lagrange identity page. David Tombe (talk) 12:47, 20 April 2010 (UTC)

This article is not intended to be exclusively about the 3D cross product

John, the reason which you gave for reverting that latest edit doesn't wash. You went to the WikiProject mathematics and you had it confirmed there that the relationship in question applies in seven dimensions. This article is about the cross product in general, even if it concentrates on the more familiar 3D cross product. If a special relationship holds in both 3D and 7D, we cannot state that this relationship is a special 3D case without also mentioning that it is a 7D case as well, because to do so would be misleading. And in that respect, the article as it now stands after your revert, is misleading, because it makes readers think that this is a relationship which is exclusively restricted to 3D. The chances are that whoever first wrote this article didn't think about the 7D case. But we have now established beyond any doubt over at the talk page of seven dimensional cross product talk page that it holds in 7D and so this article needs to be corrected in the light of that information. David Tombe (talk) 15:20, 23 April 2010 (UTC)

This article is a about the product in 3D, as the introduction says, and everything from sections 1 to 7 is written with the assumption the maths is in 3D. The reason for this is the cross product, to mathematicians, means the product in 3D. Most don't know about the product in 7D, and if they do it's given a separate name, so the 3D version can be called "The Cross Product" without ambiguity.

Towards the end it mentions the different generalisations of the cross product - there are more than just the seven dimensional one, depending on how you relax the conditions on the cross product. But the article is not also about them, or any one of them, it is just about the product in 3D.--JohnBlackburne^words_deeds 15:42, 23 April 2010 (UTC)

To settle this matter, I have added a reference template to the 7D case at the top of this article, and added the 7D case to the disambiguation page for "cross product". Brews ohare (talk) 16:31, 23 April 2010 (UTC)

John, Even if the article is only about the 3D cross product, we still need to make any general information unambiguous and not have readers thinking that a particular result is exclusive to 3D. Adding the information about 7D does not detract from the purpose of the article, whereas ommitting it can spread misinformation. So I will add the information in in brackets. Having said all that, the article shouldn't be only about the 3D cross product. It should be about the 'vector cross product' in general, with an emphasis on the more familiar 3D case. There is really no need to segregate the two articles. They can be written together coherently, pointing out the commonality and then the differences in the introduction, and then doing separate treatments of each. David Tombe (talk) 02:43, 24 April 2010 (UTC)

No, for the reasons above and earlier on this talk page. I recall that you tried to do this before, with no better arguments than now. Please drop it as there is really no point reopening this debate.--JohnBlackburne^words_deeds 08:40, 24 April 2010 (UTC)

John, You have just reverted an edit and referred me to the talk page. I'm now looking at your explanation and I can't see anything which remotely addresses the issue of why you reverted the edit. The edit in question did two unrelated things. First of all, it segregated the issue of the "Lagrange identity" into a separate sub-section from the "vector triple product". Secondly, it made an important clarification that this form of the Lagrange identity,

$|\mathbf {a} |^{2}|\mathbf {b} |^{2}-|\mathbf {a} \cdot \mathbf {b} |^{2}=|\mathbf {a} \times \mathbf {b} |^{2}\$

holds in both 3 and 7 dimensions. As the article stands now, it is misleading by its omission of any mention of the 7D case. It doesn't matter whether the article is exclusively about the 3D cross product or not. The information regarding the applicability in 7D is so closely related to the topic that its omission is positively misleading. Anybody reading the article as it stands now will easily believe that the above equation only holds in 3D. The purpose of wikipedia is to supply information and not to hold back little secrets for another day. David Tombe (talk) 14:47, 24 April 2010 (UTC)

Incorrect definition

I just restored the previous version of the definition, which is the generally accepted and most widely understood one. It is also the one given by Lounesto, on page 93. On 94 he gives an alternate definition in terms of purely vector products, so he can do algebra with it, but it is not a generally accepted or readily understood definition.--JohnBlackburne^words_deeds 20:53, 23 May 2010 (UTC)

Hi John: I'm OK with that. What I was trying for was a definition that is easily extended to n-dimensions. Can you help? Brews ohare (talk) 21:02, 23 May 2010 (UTC)

No need for that as long as the text "This article is about the cross product of two vectors in three-dimensional Euclidean space" is sitting on top of the article. DVdm (talk) 21:06, 23 May 2010 (UTC)

As has come up before the place for extending it to 7D, the only dimension it extends to, is the 7D cross product page. That's already linked in the introduction and has a section in Generalizations, so does not need anything else adding. This is a prominent article on an elementary topic, so including more content from the 7D cross product article would I think be inappropriate.--JohnBlackburne^words_deeds 21:10, 23 May 2010 (UTC)

The "cross product" of n−1 vectors in n-space is well-defined; and the properties on p.779 of the Mathematica Book, together with the "right-hand" rule

$e_{1}\times e_{2}\times \cdots \times e_{n-1}=e_{n},$

(where e_i are the unit vectors in the respective directions) should suffice. But that's a different article. The sense in which there are cross products only in dimensions 3 and 7 (with reference) might have a definition which could be extended to multilinear cross products, but I don't have a copy of any such reference with me. — Arthur Rubin (talk) 21:26, 23 May 2010 (UTC)

That's already in the article, under Multilinear algebra. The restriction to 3 and 7 dimensions is for binary products satisfying the criteria, so if you say the crosss product is a × b = c it is only is sensibly and non-trivially defined in those dimensions.--JohnBlackburne^words_deeds 21:32, 23 May 2010 (UTC)

This article includes short sections on octonion, quaternion, wedge products etc. So it is not strictly 3-D a × b. Maybe the preamble should be extended to describe its scope and its connection to other articles? The connection to cross products of n-1 vectors in n-dimensions should be brought up, for example. Brews ohare (talk) 21:47, 23 May 2010 (UTC)

It's the 67th most popular maths article, and the vast majority of those readers will I think be after the cross product as used in vector algebra, i.e. the 3D one, it's definition and basic properties. The more advanced material should come later, as it now does. Moving it to the top will just confuse many readers, as e.g. those who do maths at school won't have the math background to deal with it.--JohnBlackburne^words_deeds 22:18, 23 May 2010 (UTC)

Article topic

The article is about the cross product in 3D, i.e. what mathematicians generally understand by the name "cross product". It does not describe the generalisations: it provides links summary-style to most of them, and only the n − 1 way product in n dimensions is described at all.--JohnBlackburne^words_deeds 22:26, 23 May 2010 (UTC)

John, There is clearly a problem with this article as regards terminologies and disambiguation. The statement at the top that the article is exclusively about the binary operation in 3D is not true. As you have already said yourself, the multilinear algebra section deals with that other concept of cross product which involves multiple vectors in a single operation and extends to the corresponding multiple dimensions. It's a different subject, except in the 3D case, and it is covered in this article. The scope of this article needs to be clarified and the headings amended accordingly. David Tombe (talk) 00:47, 24 May 2010 (UTC)

John: The template claims that the article is about the cross-product in 3-space. However, it has a section on "Generalizations" which has little if anything to do with the 3-D cross product. It mentions normed division algebras, exterior algebra, and multilinear algebra including (n-1)-ary products. There are a few choices: change the template and introduction to describe the article as it actually is, or rewrite the article to fit the present description. Brews ohare (talk) 02:48, 24 May 2010 (UTC)

The article is not about the cross product in 3D "and its generalisations". It briefly mentions some generalisations. This edit was clearly making a disruptive WP:POINT in response to this remark. Please stop doing that. DVdm (talk) 09:03, 24 May 2010 (UTC)

This seems to be a rather unnecessary argument in my opinion, and it stems from the note at the top of the article which restricts the article to the 3D binary operation. I think that we are all agreed that the 3D binary operation is by far the most important and well known case. And I think that we are all agreed that the 3D binary operation should dominate the article. However, there is a section, and rightly so, which deals with the varoius ways of extending this concept to higher dimensions. The note at the top of the article is therefore inaccurate. The problem is not so much a problem with the article itself as a problem with this inaccurate and restrictive note which has been added at the top of the page. In my opinion, the article is about 'cross product'. It should deal with the 3D case first, and then discuss the extensions to higher dimensions further down. That is more or less exactly as it is at the moment. David Tombe (talk) 12:04, 24 May 2010 (UTC)

The note at the top of the article looks perfectly accurate to me. DVdm (talk) 12:30, 24 May 2010 (UTC)

It does not make sense to me

Hi, you cannot define right or left without defining up and down, and it is not in 3D space anyway, but in 2D. Can you see that? Genezistan (talk) 15:26, 25 October 2010 (UTC)

I don't follow. In n≥1 dimensions, I can define "left" to be any direction and "right" to be the opposite direction. There are issues with the cross product in any other than three dimensions, but the page discusses this. Can you rephrase your question? —Ben FrantzDale (talk) 17:46, 25 October 2010 (UTC)

I am trying to visualize what you claim to be in n≥1 dimensions, where you say you can define "left" to be any direction and "right" to be the opposite direction in 2D and I need to have up or down defined, otherwise I have a problem (rotate by 180 degrees to check).

Is that any better?

Regards, Genezistan (talk) 21:55, 25 October 2010 (UTC)

I'm missing something. In R², I'm assuming we have a coordinate system defined so a vector, x can be distinguished from −x and where −x is equal to x rotated by 180°. Without loss of generality, assume x = [−1,0]. If we assume the space is right-handed, then if x is the "left" direction, then [0,1] would be "up". Is it the assumption of handedness that you are wondering about? You are right that if we don't assume a right-handed coordinate system that we would need to define which way is up versus down in addition to left versus right. The assumption is completely standard, though, so it is usually omitted. Does that help? —Ben FrantzDale (talk) 16:59, 27 October 2010 (UTC)

Yes. Thank you very much. What do you think of the fact that in space there is symmetry, so it does not matter where you put the origo of a co-ordinate system or where you start time from?

Genezistan (talk) 17:24, 27 October 2010 (UTC)

I think you've stumbled on the distinction between a vector space and an affine space. In a vector space, there is one origin and all vectors have their tail there; in affine space, there are points and so-called bound vectors (tangent vectors in the tangent space of points). Subtracting one point from another gives a vector from one to the other. To rephrase your observation, in Euclidean space all tangent spaces are isomorphic—they look the same, so with the exception of algebraic restrictions (such as adding two points being meaningless), the space of points is isomorphic to the space of vectors, so the distiction is usually glossed over in introductory treatments. I find the affine-space treatment more pedantically-satisfying. In more-general spaces such as on manifolds, the distinction between points and tangent vectors becomes critical because points are not in general in a vector space, and each tangent space can look different (e.g., having a different metric tensor). —Ben FrantzDale (talk) 14:23, 28 October 2010 (UTC)

Also, it is possible to rotate such a 2D representation or to mirror it when you lose left or right directions again. What I am also after is the visual representation of mathematical symbols, including numbers. I do not like that in writing for instance 12345, we do not "see, because it is not there" that an individual digit is a cross product of the value of that position (power) and the number iself. I would like to have a notation that reveals such a hidden cross-product.

Genezistan (talk) 04:30, 28 October 2010 (UTC)

Or more precisely I think we should see the cross product of base x power x count explicitely for visualization purposes.

Genezistan (talk) 06:16, 28 October 2010 (UTC)

I am not sure quite what you mean. I agree that it is very important in applied math to think of quantities (numbers, vectors, tensors, etc.) as physical things, not just symbols. However, "...an individual digit is a cross product..." doesn't make any sense to me. I don't know what you mean by "hidden cross-product". —Ben FrantzDale (talk) 14:23, 28 October 2010 (UTC)

Further to that and knowing that in Phyisics where we can visualize qualities and quantities the representation of a triple cross rpoduct could be a sphere with three radiuses not necessarily connected in right angle. In fact we need four vectors and the fourth could be the unit of measure as no count makes sense without a unit of measure!

And since the three vectors are used to represnet directions in space, the fourth should autimatically be time with the difference that it has just one direction and it is positive. So the visualisation could or should be someting similar to what you get with seven cross products.

Genezistan (talk) 07:20, 28 October 2010 (UTC)

In fact the concept of spacetime is also a cross product with motion (and speed to be also considered), if I am not mistaken. What do you think?

Genezistan (talk) 07:08, 28 October 2010 (UTC)

I think you would find geometric algebra and covariance and contravariance of tensors interesting. (Be warned, it is a huge can of worms, but I think it provides the framework you are craving.) As you have seen, in the classical treatment of 3-vectors and cross products, the algebra doesn't mirror the physics: if a quantity is a cross product of vectors, it is an oriented area, and if it is a triple product it is a volume. Geometric algbra addresses this type system. Tensor variance and related concepts address the issues of how these sorts of quantites transform as you change coordinate systems. (For example, a distance gets numerically bigger if you use kilometers rather than miles, but a gradient (e.g., degrees C per distance) gets numerically smaller if you use kilometers rather than miles.) —Ben FrantzDale (talk) 14:23, 28 October 2010 (UTC)

Duplicate matrix material

I raised this here but it was deleted without a reply, so I'll raise it here. The just added material on the matrix representation duplicates what's already there, except less clearly (it's not clear what the significance of the last two lines of formulae is) and with one error (the link to Lagrange's identity is wrong - the formula is a different one with the same name).--JohnBlackburne^words_deeds 20:16, 31 May 2010 (UTC)

The added material has a plus: it's sourced. The pre-existing material is not. The Lagrange identity for vectors has a few variations. They aren't exactly the same, but are all interrelated: some involve two vectors, some three and some four. It may be too much trouble to sort them all out. Brews ohare (talk) 21:23, 31 May 2010 (UTC)

It's a different Lagrange's identity. I've seen it before, on WP I think, but you provided a link which made it clear: it's the formula

(A × B)⋅(C × D) = (A⋅C)(B⋅D) - (A⋅D)(B⋅C)

"which is sometimes known as Lagrange's identity" so the link to Lagrange's identity is incorrect.

But the matrix content is just a copy of what's already there: The matrices T_a and [a]_× are identical and they are used the same way, so the {{cn}} tag you've added is nonsensical as it's the same content you added - you've just added a perfectly good source for it.--JohnBlackburne^words_deeds 21:50, 31 May 2010 (UTC)

If the source is to be used, the notation of the source is preferable to an unsourced (invented?) notation. I am a bit surprised that the simple substitution A=C and B=D which makes the cited form of Lagrange's identity exactly that of the link happened to escape your notice. This extra generality is sometimes used in naming this identity, sometimes not.Brews ohare (talk) 00:34, 1 June 2010 (UTC)

I added the link to Binet–Cauchy identity, a more general form. Brews ohare (talk) 00:48, 1 June 2010 (UTC)

I've merged the two definitions, preferring the notation which was there before as it's more informative and there's more of it. At the same time I moved the other Lagrange's identity up, so it's clearer how it fits in, using consistent notation for that too, and saying a bit more how they're related.--JohnBlackburne^words_deeds 13:32, 2 June 2010 (UTC)

Vector quadruple product

According to [3] and [4] the vector quadruple product is (AxB)x(CxD), not (AxB).(CxD). The latter is called a "quadruple scalar product", whereas the former is called "a quadruple vector product" here. I have removed the remark added by 128.205.180.128 (talk · contribs) and which was corrected for caps by John. DVdm (talk) 14:53, 12 September 2010 (UTC)

Being introduced to it by that change I've looked at the article and have created a deletion discussion for it, hopefully to get a consensus over what if anything is the best title for an article on such an identity or identities.--JohnBlackburne^words_deeds 15:06, 12 September 2010 (UTC)

Heh. I have added a delete !vote at Wikipedia:Articles_for_deletion/Vector_quadruple_product. DVdm (talk) 15:16, 12 September 2010 (UTC)

Clifford algebra

Currently the History section contains this statement:

William Kingdon Clifford combined the algebras of Hamilton and Grassmann to produce Clifford algebra, where in the case of three-dimensional vectors the bivector produced from two vectors dualizes to a vector, thus reproducing the cross product.

Today I added the reference to Clifford's 1878 text Introduction to Dynamic which is explicit on the vector product. The above statement has no reference. Further, the allusion to Clifford Algebras does not improve the discription of cross product's history. The statement appears superfluous.Rgdboer (talk) 03:46, 9 December 2010 (UTC)

Although it does not provide references, the statement is not superfluous in my opinion. It is useful to see that mathematics, at that time, followed two paths, one of which brought the cross product. The other path produced an exterior product which, dualized in 3D, produces a cross product. There's a section in this article about "Cross product as an exterior product", and I think it is nice to have a reference to that section in the history section.

Your edit is very interesting. It seems that 3 years before Gibbs published his notes, Clifford already defined the cross product (although he did not call it cross product). Does it mean that Clifford is the father of cross product? This is weird, as I have read everywhere that Gibbs and Heaviside were the fathers of cross and dot product. Does Clifford refer to Gibbs or Heaviside, in his 1878 text?

— Paolo.dL (talk) 09:28, 9 December 2010 (UTC)

For two vectors, the vector part of their quaternion product is the cross product. So Hamilton's earlier product includes the idea. As for Clifford, he represents area, and its orientation, by a vector. This notion he may have gotten from Grassmann, but generally Clifford doesn't give references. Rgdboer (talk) 02:02, 10 December 2010 (UTC)

Who is the father of the cross product?

This sentence was inserted by Rgdboer in the history section:

• In 1878 William Kingdon Clifford published his Introduction to Dynamic which brought together many mathematical ideas. He defined the product of two vectors to have magnitude equal to the area of the parallelogram of which they are two sides, and direction perpendicular to their plane.

This sentence is very interesting, but it puzzles me. The problem is: who is the father of the cross product? I always thought that Grassmann, Hamilton and Clifford were not able to define a vector product which yielded another vector perpendicular to their plane (what we now call the cross product). Here's why they, as far as I know, were not the fathers:

1. The quaternion product was defined by Hamilton before the cross product, but quaternions are 4-element vectors. Although the vector part of the quaternion product, which involves only 3 elements, is what now we would call a cross product, Hamilton failed to see it. So, Hamilton is not the father of the cross product.
2. The exterior product (or wedge procuct) was defined even earlier by Grassmann. In R³ it is the same as a cross product as far as its "scalar components" are concerned. However, its "vector components" are bivectors, not unit vectors. And a bivector, as far as I know, was described and interpreted as a directed parallelogram (a 2-D surface) with an area, not as a directed line segment with a length. I guess that this also means that, if a vector is measured in meters, the exterior product is in square meters. So, Grassmann was not the father.
3. As far as I know, the Clifford product is similar to, but even more complex than the exterior product, as it takes into account an additional argument, called Q. So, Clifford was going away from the cross product, rather than toward its discovery.

However, the above mentioned sentence inserted by Rgdboer in the history section seems to say something different.

Question: Did Clifford, in 1878, 3 years before the publication of the notes in which Gibbs described for the first time its cross product, really define a "product of two vectors" which returned another vector (i.e. a directed line segment) in the same vector space? This would not be an exterior product or a Clifford product, but a vector product identical to what we now call a cross product! The answer to this question is crucial.

— Paolo.dL (talk) 15:58, 10 December 2010 (UTC)

I would not be surprised if he did (though the mentioned source seems not to exist) – the cross product can be derived from the exterior product of Grassmann within geometric algebra by taking the straigtforward dual of the bivector result of the exterior product, and this is only possible with Clifford's geometric algebra, newly discovered in 1878. Geometric algebra includes both Grassmann's algebra and quaternions so can seem more complex, though it can also used to define both from very simple axioms. Gibbs' cross product though was derived from the quite different quaternion product, by taking just the imaginary part, ignoring the insights available with the full quaternion algebra, seemingly unaware of Clifford's deeper discoveries. But that paragraph does seem out of place, as Clifford is mentioned later.--JohnBlackburne^words_deeds 16:35, 10 December 2010 (UTC)

Mentioned later, but for a different reason. The inserted sentence does not say that you need a dual to obtain a vector. It seems to say that the "product of two vectors" defined by Clifford in 1878 is already a vector (not a bivector). Paolo.dL (talk) 16:57, 10 December 2010 (UTC)