Exercise 3.80

Solution to Exercise 3.80.
Knowledge consists in understanding the evidence that establishes the fact, not in the belief that it is a fact..
—Charles T. Sprading (?–1960)

Exercise 3.80. Find a matrix of the reflection R_l where (a) l[1, –1, 0] (b) l[0, 1, –4], then for each reflection find the image of (4, 4, 1) and

(a) l[1, –1, 0] Two Methods are given.
Method 1.

We first find the point of intersection of h[0, 1, 0] and line l by applying Proposition 3.2,

Hence, u₃ = 0. Thus, the point of intersection is the origin O(0, 0, 1). The measure of the angle between lines h and l is

Hence, by Proposition 3.15,

The image of (4, 4, 1) is (4, 4, 1) since

The image of is since

Method 2.

Use Proposition 3.6 to find the matrix of a direct isometry T that maps h[0, 1, 0] to l[1, –1, 0] .

implies

Hence, and b = 0. For a matrix that translates line h to line l, let a = 0,

Hence, R_l is defined by

The image of (4, 4, 1) is (4, 4, 1) since

The image of is since

(b) l[0, 1, –4] one method given.

Since l[0, 1, –4] is parallel to h[0, 1, 0], a direct isometry that maps h to l is a translation determined by the vector (0, 4, 0). Hence, by Proposition 3.15,

The image of (4, 4, 1) is (4, 4, 1) since

The image of is since

Solutions for Chapter 3