Let v = (a, b, c) and let , , and be the angles between v and the positive x-axis, the positive y-axis, and

Let v = (a, b, c) and let α, β, and γ be the angles between v and the positive x-axis, the positive y-axis, and the positive z-axis, respectively (see figure).

a. Prove that cos² α + cos² β + cos² γ = 1.

b. Find a vector that makes a 45° angle with i and j. What angle does it make with k?

c. Find a vector that makes a 60° angle with i and j. What angle does it make with k?

d. Is there a vector that makes a 30° angle with i and j? Explain.

e. Find a vector v such that α = β = γ. What is the angle?