 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/c/solution

M x = x - \hat{n} (\hat{n}^{T} x) = x - (\hat{n} \cdot x) \hat{n} .

Since

\hat{n}

is a unit vector, the subtracted term is the orthogonal projection of

x

along

\hat{n}

. Hence $M$ is the orthogonal projection onto a plane through the origin with normal vector $\hat{n}$ . It fixes every vector perpendicular to

\hat{n}

and sends every scalar multiple of

\hat{n}

to zero. In particular

\hat{n} \cdot M x = 0

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