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The above transformation can also be written in matrix form. That is, if the
point is represented by a size 2 column vector, then the new vector is related
to the old one through the following transformation:

x ′ cos( )θ − sin( )θ    x   x
  =     = ()θR   (9.7)
′ y
   sin( )θ cos( )θ     
y
y
The convention for the sign of the angle is the same as that used in Chapter 6,
namely that it is measured positive when in the counterclockwise direction.

Preparatory Exercises

Using the above form for the rotation matrix, verify the following properties:
Pb. 9.4 Its determinant is equal to 1.
–1
Pb. 9.5 R(–θ) = [R(θ)] = [R(θ)] T
Pb. 9.6 R(θ ) ∗ R(θ ) = R(θ + θ ) = R(θ ) ∗ R(θ )
2
1
1
1
2
2
2
2
2
Pb. 9.7 (x′) + (y′) = x + y 2
Pb. 9.8 Show that P = R(θ = π). Also show that there is no rotation that can
reproduce P or P .
y
x
In-Class Exercises
Pb. 9.9 Find the coordinates of the image of the point (x, y) obtained by
reflection about the line y = x. Test your results using MATLAB.
Pb. 9.10 Find the transformation matrix corresponding to a rotation of
–π/3, followed by an inversion around the origin. Solve the problem in two
different ways.
Pb. 9.11 By what angle should you rotate the trapezoid so that point (6, 1)
of the trapezoid of Example 9.1 is now on the y-axis?

9.1.4 Scaling
If the x-coordinate of each point in the plane is multiplied by a positive con-
stant s , then the effect of this transformation is to expand or compress each
x
plane figure in the x-direction. If 0 < s < 1, the result is a compression; and if
x
s > 1, the result is an expansion. The same can also be done along the y-axis.
x
This class of transformations is called scaling.
© 2001 by CRC Press LLC