Talk:Dot product
From Academic Kids
This needs some major merging and clarification of terminology with the article inner product space. For example, this article never defines what a dot product is, esp. over an arbitrary vector space. Then it defines the dot product in terms of the "angle" between two vectors, which is never defined. This is circular definition. The typical way is to define the cosine between two vectors in terms of the inner product. Then all of what follows here is just special case when V is R^n with usual dot product. Personally, I think there could be two articles, the inner product space one giving the abstract algebraic formulation, and a more concrete geometric one focusing just on R^n or C^n, aimed more at people who might just use it for calculus or physics classes. Revolver 00:00, 1 Apr 2004 (UTC)
A quote from the main article between the horizontal rules:...
Properties
The definition has the following consequences:
- the dot product is commutative, i.e. a·b = b·a.
- two non-zero vectors a and b are perpendicular if and only if a·b = 0
- the dot product is bilinear, i.e. a·(rb + c) = r (a·b) + (a·c)
From these it follows directly that the dot product of two vectors a = [a1 a2 a3] and b = [b1 b2 b3] given in coordinates can be computed particularly easily:
- a·b = a1b1 + a2b2 + a3b3
I'm afraid that I don't see how "it follows directly".
It can be easily shown with 2D vectors that this is true using "cos(A-B)=cosA.cosB+sinA.sinB", but I'm not sure how to extend this to 3D (or more D) vectors.
Is it worth expanding the article to show this/these derivation(s)? -- SGBailey 21:56, 2003 Nov 16 (UTC)
Given a basis of perpendicular unit vectors, it does follow at once.
Charles Matthews 09:53, 19 Nov 2003 (UTC)
- I accept that it works. I just don't see how the three consequences "directly" cause the ax.bx+ay.by+az.cz construct. I used to know this stuff 30 years ago - sigh. -- SGBailey 2003-11-18
I dislike the first couple of prargraphs now. They are diving into too much depth without giving a general overview first. However I'm not able to edit it without losing much of the content of those paras. The equations for the dot product and its description want to come before the stuff about vector spaces and fields. -- SGB 2004-03-24
I would like to point out that most of this article is related to *real* vector spaces, when the field is not the reals, things are different. For example when the characteristic is non null, for example Z/5Z, then there are vectors in the 2-dimensional vector space (Z/5Z)² which are such that <x,x>=0, for example x=(1,2), without having <x,y>=0 for all y. And there no chance to define the notion of norm or angles between lines (there are 6 of them). All the things about angles should be left to the real case only. -- Christian Mercat 2004-04-01
I agree with some of the concerns mentioned above. First, as far as I know, the term dot product is only used in real vector spaces. Over an arbitrary field, the common used terminology is bilinear form. Even over the complex numbers, dot product sounds a little too conversational; in this case I would instead say inner product, and refer to it as a Hermitian form.
Apart from terminology, the article currently features considerable confusion between the real and arbitrary field cases. Dmharvey Missing image
User_dmharvey_sig.png
Image:User_dmharvey_sig.png
Talk 21:58, 3 Jun 2005 (UTC)
