Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2002 / Paper 1 / 1 / a

Write $\mathbf{r}=\mu \mathbf{n}+\mathbf{v}$. Since $\mathbf{n}$ is a unit vector, the equation becomes $\mathbf{v}\cdot \mathbf{n}=0$. Thus $\mathbf{v}$ ranges over the two-dimensional orthogonal complement of $\mathbf{n}$: the locus is an affine hyperplane through $\mu \mathbf{n}$, with unit normal $\mathbf{n}$. Conversely, every such vector satisfies the equation. Its perpendicular distance from the origin is $\boxed{|\mu |}$; $\mu$ is the signed distance in the chosen normal direction.