|a| = (16 + 16 + 16) = 4  3 ,  |b| = (16 + 16)      =  4  2  
au  =  (4, 4, 4)T/(4 3) ,  bu  =  (4, 0, 4)T/(4 2  )
au · bu = (16 + 16)/( (43)(42) )  
   = 2/(3  2 )  = 2 /3 = cos θ
cos θ =  0.81649, θ  =  35.26°

A More Difficult Problem

 

The nasty math in the previous exercise is not the real purpose of all this. The goal is to illustrate the formula a_u · b_u  =  cos θ ,   which is important in every part of 3D graphics. It is worth another example.

The figure shows two vectors, represented by:

f   =   (4,3,2)^T

g   =   (-1,4,4)^T

Rotate the figure to get a better sense of the angle. What you would like to do is to lay a sheet of paper across the two vectors, trace them onto the paper, and then measure the angle with a protractor. But this is hard to do with a computer screen.