Page 331 - Matrix Analysis & Applied Linear Algebra

5.6 Unitary and Orthogonal Matrices 327

it is evident (see Figure 5.6.4) that the third coordinates are unaffected—i.e.,
v 3 = u 3 . To see how the xy-coordinates of u and v are related, consider the
orthogonal projections

u p =(u 1 ,u 2 , 0) and v p =(v 1 ,v 2 , 0)

of u and v onto the xy-plane.
z

u = (u , u , u ) v = (v , v , v )
1 2 3 1 2 3
θ

y

u = (u , u , 0) θ v = (v , v , 0)
p 1 2 p 1 2

x
Figure 5.6.4
It’s apparent from Figure 5.6.4 that the problem has been reduced to rotation
in the xy-plane, and we already know how to do this. Combining (5.6.13) with
the fact that v 3 = u 3 produces the equation
     
cos θ − sin θ 0
v 1 u 1
  =  sin θ cos θ 0    ,
v 2 u 2
0 0 1
v 3 u 3
so
 
cos θ − sin θ 0
P z =  sin θ cos θ 0 
0 0 1
3
is the matrix that rotates vectors in counterclockwise around the z-axis
through an angle θ. It is easy to verify that P z is an orthogonal matrix and
that P −1 = P T rotates vectors clockwise around the z-axis.
z z
By using similar techniques, it is possible to derive orthogonal matrices that
rotate vectors around the x-axis or around the y-axis. Below is a summary of
3
these rotations in .