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

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- Solution 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} .

Their cross product is the perpendicular unit vector

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

is an orthogonal matrix.

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