(a) Prove that, for all ,,, is a proper orthogonal matrix. (b) Write down a formula for Q1. Remark. It can be shown that every

(a) Prove that, for all θ,ϕ,ψ,

is a proper orthogonal matrix.

(b) Write down a formula for Q−1. Remark. It can be shown that every proper orthogonal matrix can be parameterized in this manner; θ,ϕ,ψ are known as the Euler angles, and play an important role in applications in mechanics and geometry.

Q= || cos - cos cos sin sin cos sin sin cos sin cos sin sin cos + cos sin sin + cos - sin cos cos sin cos cos sin sin sin cos cos