 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/17/f/solution

Q = 010100001, Q x = (x_{2}, x_{1}, x_{3})^{T} .

Applying

P

from part (e) then gives

y = PQ x = (- x_{1}, - x_{2}, x_{3})^{T}, PQ = diag (- 1, - 1, 1) .

Q

reflects in the plane

x_{1} - x_{2} = 0

q

p

, so the two reflecting planes meet at a right angle along the

x_{3}

axis. Their composition is a rotation through $π$ about the $x_{3}$ axis, as is also immediate from the displayed rotation matrix.

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