Talk:Cross product

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consider adding some references--Cronholm144 08:26, 20 May 2007 (UTC) Pictures are very good. Needs more applications, particularly in integral calculus (which is currently not even mentioned). Silly rabbit 06:03, 4 June 2007 (UTC)

1 Cross product in N dimensions
2 My formula vs yours
3 Edits concerning parallelipiped and Lagrange's formula
4 Explaination of the right hand rule
5 Simple formula
6 Simplest examples
7 Disambiguation: cross products in topology
8 MathML versus HTML
9 Pseudovectors
- 9.1 A few comments
10 Magnitude
11 Conventional vector algebra
12 changes needed
13 Did Joseph Louis Lagrange know the cross product before it was invented? (Part 1)
14 Asking questions vs giving answers
15 Did Joseph Louis Lagrange know the cross product before it was invented? (Part 2)

[edit] 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

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] "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)

[edit] My formula vs yours

Am I missing something, or is this wrong?

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

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

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

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

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

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

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

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

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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

[edit] 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)

[edit] 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)

[edit] A few comments

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

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

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

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

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

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

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

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

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

[edit] Magnitude

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

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

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

EikwaR 22:07, 20 May 2007 (UTC)

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

[edit] 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?

Conventional
Gibbs
Classic
Basic
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)

[edit] 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)

[edit] 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)

[edit] 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.

My use of "must" was conditional, not imperative. I never stated nor implied, in any common understanding of the English language, that you are compelled to answer your questions; I only said that Wikipedia needs champions to get the hard work done.
I could hardly have been more explicit in giving you credit when I said "You started down the right path…"; the fact that my words lay dormant on the talk page for so long before they were added to the article underlines my point that our great need is for those who are willing to do the work of researching and editing.
Perhaps you do not have a DjVu plugin installed with your browser, and clearly the standard practice of the archive is unfamiliar to you; but on the left side of the item page is a listing of four different formats (DjVu, PDF, TXT, Flip book) and an FTP page, as well as a link to "Help reading texts". The DjVu version is ¹⁄₄ 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)

[edit] 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)

[edit] 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, <http://www.archive.org/details/117714283>

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-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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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 (spatial), 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)