# 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).

http://mathforum.org/library/drmath/view/53928.html

http://www.physics.orst.edu/bridge/mathml/dot+cross.xhtml

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