Some books actually use as the definition of the dot product. Another definition is .
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 . Similarly I’ll let the vector u have two components (, ) and v be (, ) .
I want to find cos(x), which is:
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).