vi 0 0 Rotation Matrices 4.
C Suppose that ↵ R. We let 2 vi C 2 2 vi 0 0 R↵ : R R !
be the4. function defined as follows: C Any vector in the plane can be written in polar coordinates as r(cos(✓), sin(✓)) vi C c) where r 0and✓ R. For any such vector, we define 0 A 2 R↵ r(cos(✓), sin(✓)) = r(cos(✓ + ↵), sin(✓ + ↵)) Notice that the function R↵ doesn’t change the norms of vectors (the num- c) 0
A ber r), it just a↵ects their direction, which is measured by the unit circle coordinate. We call the function R↵ rotation of the plane by angle ↵. 0 + v-b + 0 0 + + vi vi 0 0 4. 0 + v-b + 0 0 + + vi C vi C vi 0 0 4. c) C 0 A vi C c) 0 A
If ↵ > 0, then R↵ rotates the plane counterclockwise by an angle of ↵. If 0 + v-b + ↵ < 0, then R is a clockwise0 rotation by an angle of ↵ . The rotation does ↵ 0 + + | | vi not a↵ect the origin in the plane. That is, R↵(0, 0) = (0, 0) always, no matter which number ↵ is. 258 0 + v-b + 0 0 + + vi Examples.
π ◦ • R π is the function that rotates the plane by an angle of , or 90 . 2 2 π Because > 0, it is a counterclockwise rotation. Thus, R π (1, 1) is the point 2 2 in the plane that we obtain by rotating (1, 1) counterclockwise by an angle π of 2 .
I — (H) a ‘ S
π • Because − < 0, R π is a clockwise rotation. R π (1, 1) is the point 2 − 2 − 2 π
in the plane obtained by rotating (1, 1) clockwise by an angle of 2 . (I,’)
2 2 • The function R0 : R → R rotates the plane by an angle of 0. That is, it doesn’t rotate the plane at all. It’s just the identity function for the plane. 259 • Below is the picture of a shape in the plane. It’s a triangle, and we’ll call this subset of the plane D.
D
R π (D) is the set in the plane obtained by rotating D counterclockwise by 2 π π an angle of 2 . (It’s counterclockwise because 2 > 0.)
π R π (D) is D rotated clockwise by an angle of . − 4 4
260 • Let’s rotate the vector (a, 0), where a ≥ 0. This is a point on the x-axis whose norm equals a.
I (i,o) (ao’):o.(l,o)
We can write this vector in polar coordinates as a(1, 0), or equivalently, as a( cos(0), sin(0) ). Now we(o’i)v: can rotate the vector (a, 0) by an angle α. That’s
the vector Rα(a, 0), which by the formula from the beginning of this chapter is (o’i)v:
Rα(a, 0) = Rα a( cos(0), sin(0) )
= a( cos(0 + α), sin(0(o’i)v: + α))
‘° (o’z)
0 = a( cos(α), sin(α)) (o’i)v:
‘° (o’z) 0
‘° (o’z) 0
‘° (o’z) 0 (o’i)v:
• In this example, we’ll rotate a vector (0, b), where b ≥ 0. This is a vector whose norm equals b, and that points straight up. In polar coordinates,