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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.

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Thanks HallsOfIvy, I'll try that.

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[tex]x_{a}=a\cos(\theta_{a})[/tex]

[tex]y_{a}=a\sin(\theta_{a})[/tex]

[tex]x_{b}=b\cos(\theta_{b})[/tex]

[tex]x_{b}=b\sin(\theta_{b})[/tex]

whereby we get:

[tex]x_{a}x_{b}+y_{a}y_{b}=ab\cos(\theta_{a}-\theta_{b})[/tex]

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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.

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

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

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Strictly speaking the term "dot product" is only

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

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

And the cross product is only defined for R

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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.

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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').

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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.

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