Talk:Cross product

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1 removed formula
2 question
3 my formula vs yours
4 Crossproduct dimension
5 Edits concerning parallel piped and lagrange's formula
6 explaination of the right hand rule
7 Simple formula
8 Simplest examples
9 disambiguation: cross products in topology

[edit] removed formula

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)

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] question

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] 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)

[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] Edits concerning parallel piped 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)

Its 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

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