Page 330 - Matrix Analysis & Applied Linear Algebra

326 Chapter 5 Norms, Inner Products, and Orthogonality

2
Now consider rotation, and begin with a basic problem in . If a nonzero
vector u =(u 1 ,u 2 )is rotated counterclockwise through an angle θ to produce
v =(v 1 ,v 2 ), how are the coordinates of v related to the coordinates of u?To
answer this question, refer to Figure 5.6.3, and use the fact that u = ν = v
(rotation is an isometry) together with some elementary trigonometry to obtain

v 1 = ν cos(φ + θ)= ν(cos θ cos φ − sin θ sin φ),
(5.6.12)
v 2 = ν sin(φ + θ)= ν(sin θ cos φ + cos θ sin φ).

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

Figure 5.6.3
Substituting cos φ = u 1 /ν and sin φ = u 2 /ν into (5.6.12) yields

v 1 = (cos θ)u 1 − (sin θ)u 2 , v 1 cos θ − sin θ u 1
or = . (5.6.13)
v 2 = (sin θ)u 1 + (cos θ)u 2 , v 2 sin θ cos θ u 2
In other words, v = Pu, where P is the rotator (rotation operator)

cos θ − sin θ
P = . (5.6.14)
sin θ cos θ
T
Notice that P is an orthogonal matrix because P P = I. This means that if
T
v = Pu, then u = P v, and hence P T is also a rotator, but in the opposite
direction of that associated with P. That is, P T is the rotator associated with
the angle −θ. This is confirmed by the fact that if θ is replaced by −θ in
(5.6.14), then P T is produced.
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Rotating vectors in around any one of the coordinate axes is similar.
For example, consider rotation around the z-axis. Suppose that v =(v 1 ,v 2 ,v 3 )
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is obtained by rotating u =(u 1 ,u 2 ,u 3 ) counterclockwise through an angle
θ around the z-axis. Just as before, the goal is to determine the relationship
between the coordinates of u and v. Since we are rotating around the z-axis,

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This is from the perspective of looking down the z-axis onto the xy-plane.