 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2/17/b/solution

The two prescribed directions have dot product zero. Normalize them to

q_{1} = \frac{( 1 , 1 , 1 ) ^{T}}{3}, q_{2} = \frac{( 0 , - 1 , 1 ) ^{T}}{2} .

q_{3} = (2, - 1, - 1)^{T} / 6

. Using these as columns gives

Q = 1/ 3 1/ 3 1/ 3 0 - 1/ 2 1/ 2 2/ 6 - 1/ 6 - 1/ 6 .

The columns form an orthonormal basis, so

Q^{T} Q = I

and

Q

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