= Solution
Apply <matrix multiplication> to obtain
$$
M\mathbf x=\mathbf x-\hat{\mathbf n}(\hat{\mathbf n}^T\mathbf x)=\mathbf x-(\hat{\mathbf n}\cdot\mathbf x)\hat{\mathbf n}.
$$
Since $\hat{\mathbf n}$ is a <unit vector>, the subtracted term is the <orthogonal projection> of $\mathbf x$ along $\hat{\mathbf n}$. Hence \b[$M$ is the <orthogonal projection onto a plane> through the origin with <normal vector> $\hat{\mathbf n}$]. It fixes every <vector> perpendicular to $\hat{\mathbf n}$ and sends every <scalar multiple> of $\hat{\mathbf n}$ to zero. In particular $\hat{\mathbf n}\cdot M\mathbf x=0$.
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