Deriving the Dot Product

Some books actually use |U||V|\cos(\theta) as the definition of the dot product. Another definition is u \cdot v =x_u x _v +y_u y_v.

The idea is to find the angle between two vectors. At way to do this is to look at the angles made with the x-axis. We want to know the difference between the two angles, which I’ll call \theta_u, \theta_v. Similarly I’ll let the vector u have two components (x_u, y_u) and v be (x_v, y_v) .

I want to find cos(x), which is:

\displaystyle \cos(\theta_u -\theta_v)  = \cos \theta_u \cos \theta_v + \sin \theta_u \sin \theta_v  = (\frac{x_u }{ |u|}) (\frac{x_v }{ |v|}) + (\frac{y_u} {|u|}) (\frac{y_v }{ |v|})

Very simple, it turns out, when you look at it the right way.

Make sense? Now, why don’t you try to derive the same result for
3-dimensional vectors? If you’re slick, you can actually use the
2-dimensional result (hint: there’s a plane that contains the two
vectors and the origin).


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