Solution to Exercise 3.62.
Science is simply common sense at its best-that is, rigidly accurate in observation, and merciless to fallacy in logic.
—Thomas H. Huxley (1825–1895)

Exercise 3.62. Let the point of intersection of the lines l[–1, 5, 1] and m[3, –2, 4] be the center of a rotation that maps l to m. Find the matrix of a rotation that maps l to m.

 

Two Methods are given.

Method 1.

        We first find the point of intersection by applying Proposition 3.2.

 

Hence, 22u₁ + 7u₂ – 13u₃ = 0; that is, (–22/13)u₁ + (–7/13)u₂ + u₃ = 0. Thus, the point of intersection C is (–22/13, –7/13, 1). The measure of the angle between lines l and m is

 

Hence, by Proposition 3.14,

 

 

Method 2.

        We first find the point of intersection by applying Proposition 3.2.

 

Hence, 22u₁ + 7u₂ – 13u₃ = 0; that is, (–22/13)u₁ + (–7/13)u₂ + u₃ = 0. Thus, the point of intersection C is (–22/13, –7/13, 1).

        We apply Proposition 3.6; there is a nonzero real number k such that kmA = l where A is the matrix of an affine rotation.

 

 

 

 

Solve the system of equations to obtain   and  Thus, the matrix that rotates line l to line m with center (–22/13, –7/13, 1) is