Chapter 14.2, Problem 91E

Vectors r and r′ for lines

1. a. If r(t) = 〈at, bt, ct〉 with 〈a, b, c〉 ≠ 〈0, 0, 0〉, show that the angle between r and r′ is constant for all t > 0.
2. b. If r(t) = 〈x₀ + at, y₀ + bt, z₀ + ct〉, where x₀, y₀, and z₀ are not all zero, show that the angle between r and r′ varies with t.
3. c. Explain the results of parts (a) and (b) geometrically.