Solution to Exercise 3.81.
Out of clutter, find simplicity.
From discord, find harmony.
In the middle of difficulty lies opportunity
—
Albert Einstein (1879–1955)
Exercise 3.81. Find a matrix of the reflection Rl where and find the image of (2, 8, 1), (4, 4, 1) and
(10, 7, 1).
Three Methods are given.
Method 1 and Method 2.
We first find the point of intersection of h[0, 1, 0] and line l by applying Proposition 3.2.
Hence, Thus, the point of intersection C is
The measure of the angle between lines h and l is
We apply Proposition 3.15. Note there are many ways for determining a matrix for direct isometry that may be used to find the reflection matrix. Method 1 is to use the rotation matrix that maps h to l with C as the center of rotation:
Method 2 is to map h to l by rotating about the origin, then translating the origin to C:
Note that the second approach
requires fewer matrices to multiply; therefore, we will use the second approach
in all future exercises.
The image of (2, 8, 1) is
The image of (4, 4, 1) is
The image of (10, 7, 1) is
Method 3.
We apply
Proposition 3.6 to find
the matrix of a direct isometry T that
maps h[0, 1, 0] to There is a nonzero
real number k such that klT = h.
implies
Hence, k = 1/2,
and
For a matrix for a direct isometry that maps line h to line l, let a = 0 and b = 1,
Hence, Rl is defined by
The image of (2, 8, 1) is
The image of (4, 4, 1) is
The image of (10, 7, 1) is
