Deriving the algebraic definition the dot product

[autodidude]

Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines?

 

[HallsofIvy]

What exactly do you mean by "the geometric definition"? The simplest "geometric" definition I know is that the dot product of vectors u and v is the product of the length of u and the length of the projection of v on u, but I suspect that you are thinking of "length of u times length of v times cosine of the angle between them". Since that has "cosine" already in it, if not using the "cosine law" per se, you certainly will need to use something that has a "cosine" in it.

From the definition "length of u times the length of the projection of v on u", I would start by setting up a coordinate system in which the positive x-axis lies along vector u. Then u has components <a, 0>. Taking <b, c> to be the components of v, the "projection of v on u" is just <b, 0> so the dot product is ab= ab+ c0, the usual component formula for the dot product in this case. Get the general formula by rotating axes.

 

[autodidude]

Thanks HallsOfIvy, I'll try that.

 

[arildno]

Well, in the 2-D case, given vectors "A", "B", with $(x_{a},y_{a})$, $(x_{b},y_{b})$, lengths a and b, respectively you may think in terms of right-angled triangles, and set up:
$$x_{a}=a\cos(\theta_{a})$$
$$y_{a}=a\sin(\theta_{a})$$
$$x_{b}=b\cos(\theta_{b})$$
$$x_{b}=b\sin(\theta_{b})$$
whereby we get:
$$x_{a}x_{b}+y_{a}y_{b}=ab\cos(\theta_{a}-\theta_{b})$$

 

[MendelCyprys]

First define the dot product for A and B to be the product of their magnitudes and the cosine of the angle between them.
We can see geometrically that A.(B + C) = A.B + A.C (think about the component of B and C along A), and therefore (A + B).(C + D) = A.C + A.D + B.C + B.D.
Choosing perpendicular axis, every vector can be written in terms of components, so A = a_1*i + a_2*j and B = b_1*i + b_2*j.
Therefore A.B = a_1*b_1*i.i + a_2*b_2*j.j + a_1*b_2*i.j + a_2*b_1*j.i.
Because i and j are unit vectors and perpendicular, i.i = 1, j.j = 1, i.j = 0, j.i = 0.
So we are left with A.B = a_1*b_1 + a_2*b_2.

 

[jedishrfu]

Also one thing to be aware of is that the algebraic definition for vector dot and cross products only work when you have your vectors defined in a Euclidean space like our old favorite x,y,z or i,j,k.

and of course here's more info on it from wikipedia:

http://en.wikipedia.org/wiki/Vector_dot_product

 

[HallsofIvy]

Also one thing to be aware of is that the algebraic definition for vector dot and cross products only work when you have your vectors defined in a Euclidean space like our old favorite x,y,z or i,j,k.

and of course here's more info on it from wikipedia:

http://en.wikipedia.org/wiki/Vector_dot_product

Strictly speaking the term "dot product" is only used Euclidean space. In other vector spaces the term is "inner product". Of course, any n-dimensional vector space is isomorphic to Rⁿ so the two work out to be "essentially" the same.

And the cross product is only defined for R³, not general Euclidean spaces.

 

[jedishrfu]

Its also interesting to note the beauty of the inner and outer products. The inner product is a projection of one vector on another and the outer product is the non-projectable component of one vector on another.

Since the projection interpretation is valid for either vector then the resultant vector must be perpendicular to both and thus it becomes normal to the plane containing the two vectors.

 

[autodidude]

Couldn't we just always work a coordinate system where the x-axis is parallel to the x-component of one vector?
Or is that what you mean when you said 'get the general formula by rotating axes?'. And would the formula only work in two dimensions? [strike]If so, what about higher dimensions? [/strike]

EDIT: Woops, didn't see the new posts! SO for rotating the axes, would it much more complicated in 3D?

Thanks arildno, when you times the multiply the components and add them up, that just comes from the definition of dot product that HallsOfIvy mentioned right? ('length of u times the length of the projection of v on u').

 

[lavinia]

Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines?

Project the vector A onto the line through B. Multiply the length of this projection by the length of B. By similar triangles this is the same as projecting the vector B onto the line through A and multiplying the length of the projection by the length of A.

Call these products the dot product of A and B.

Either product divided by AB gives you the same number which depends only on the lines and not on the particular vectors.

Call this number the cosine of the angle between the vectors.

P.S. In Euclidean geometry the law of similar triangles which is what is used here is logically equivalent to the Pythagorean Theorem so using the Law of Cosines is really no different than what is posted here.