 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/5/d/i/solution

A product of orthogonal matrices is orthogonal. In fact,

R_{1}^{T} R_{1} = R_{y}^{T} R_{z}^{T} R_{z} R_{y} = R_{y}^{T} R_{y} = I .

Thus

R_{1} is orthogonal for every θ, ϕ .

Its determinant is also one, since each factor has determinant one, so it remains a proper rotation matrix.

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