Exercise 3.38

Solution to Exercise 3.38.
I think there's no way they should have to teach [math] now. We have computers. We no longer need to know why 3x = 2y/4.
Rosie O'Donnell Newsweek (April 9, 2001)
(How would you respond?)

Exercise 3.38. An affine transformation maps X(5, 0, 1) to X'(4, 6, 1) and Y(0, 0, 1) to Y'(1, 2, 1).
(a) Show d(X, Y) = d(X', Y') and show the transformation may not be an isometry.

First, we show the distance relationship. and Hence, d(X, Y) = d(X', Y'). Next, we find an affine transformation of the Euclidean plane that preserves this distance but is not an isometry.

implies

Consider that satisfies both of the above conditions. Since det(A) = 2, by the Corollary to Proposition 3.7, A is not the matrix of an isometry.

(b) Find a direct isometry for the transformation.

implies

Hence, a direct isometry is

implies

Hence, an indirect isometry is

(d) Find the image of Z(3, 10, 1) for the isometries you obtained in parts (b) and (c).

(b)

(c)