Solution to Exercise 3.61.
Learning, experimenting, observing, try not to stay on the surface of facts. Do not become the archivists of facts. Try to penetrate to the secret of their occurrence, persistently search for the laws which govern them.
—Ivan Pavlov (1849–1936)

Exercise 3.61. Let the point of intersection of the lines l[1, –1, 0] and m[1, 1, –2] 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, 2u₁ + 2u₂ + 2u₃ = 0; that is, u₁ + u₂ + u₃ = 0. Thus, the point of intersection C is (1, 1, 1).

By the definition of the measure of the angle between two lines,  since (1)(1) + (–1)(1) = 0. Thus, by Proposition 3.14,

 

 

Method 2.

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

 

Hence, 2u₁ + 2u₂ + 2u₃ = 0; that is, u₁ + u₂ + u₃ = 0. Thus, the point of intersection C is (1, 1, 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 k = 1,  and  Thus, the matrix that rotates line l to line m with center (1, 1, 1) is