Exercise 3.83

Solution to Exercise 3.83.
I seem to have been only like a boy playing on the seashore and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary whilst the great ocean of truth lay all undiscovered before me.
—Sir Isaac Newton (1642–1727)

Exercise 3.83. Find a matrix of the reflection that maps the line l[2, 3, –1] to m[2, 3, 5].

Two Methods are given.

Method 1.

First, note that the axis of reflection must be p[2, 3, 2]. We first find the point of intersection of line h and p, by applying Proposition 3.2.

Hence, Thus, the point of intersection C is (–1, 0, 1). The measure of the angle between lines h and p is

Note:

and

Hence, by Proposition 3.15,

We check the solution by applying Proposition 3.6.

The result checks when k = 1.

Method 2.

First, note that the axis of reflection must be p[2, 3, 2]. We apply Proposition 3.6 to find the matrix of a direct isometry T that maps h[0, 1, 0] to p[2, 3, 2]. There is a nonzero real number k such that kpT = h.

implies

Solve this system with the fact that det(T) = 1 to obtain, and For a matrix of a direct isometry that maps line h to line l, let a = 2 and b = –2,

Hence, R_p is defined by

We check the solution by applying Proposition 3.6.

The result checks when k = 1.

Solutions for Chapter 3