Page 332 - Matrix Analysis & Applied Linear Algebra

328 Chapter 5 Norms, Inner Products, and Orthogonality

3
Rotations in R
3
A vector u ∈ can be rotated counterclockwise through an angle θ
around a coordinate axis by means of a multiplication P u in which
P is an appropriate orthogonal matrix as described below.
Rotation around the x-Axis
z
 
1 0 0
P x =  0 cos θ − sin θ 
0 sin θ cos θ y
θ
x
Rotation around the y-Axis
z
 
cos θ 0 sin θ
P y =  0 1 0  θ
− sin θ 0 cos θ y

x
Rotation around the z-Axis
z
 
cos θ − sin θ 0
θ
P z =  sin θ cos θ 0 
0 0 1 y
x
Note: The minus sign appears above the diagonal in P x and P z , but
below the diagonal in P y . This is not a mistake—it’s due to the orien-
tation of the positive x-axis with respect to the yz-plane.

Example 5.6.4
3-D Rotational Coordinates. Suppose that three counterclockwise rotations
are performed on the three-dimensional solid shown in Figure 5.6.5. First rotate
the solid in View (a) 90 ◦ around the x-axis to obtain the orientation shown
in View (b). Then rotate View (b) 45 ◦ around the y-axis to produce View (c)
and, finally, rotate View (c) 60 ◦ around the z-axis to end up with View (d).
You can follow the process by watching how the notch, the vertex v, and the
lighter shaded face move.