Past exam of the mathematical methods I course of the University of Oxford / 2016 / 3 / Solution

The scalar product $\mathbf{r}\cdot \hat{\mathbf{n}}=d$ fixes the component of the position vector along the given unit normal vector. Thus the locus is a plane normal to $\hat{\mathbf{n}}$, through $d\hat{\mathbf{n}}$. Its signed distance from the origin along that normal is $d$, and its perpendicular distance from the origin is $|d|$. Any displacement lying within the plane is orthogonal to $\hat{\mathbf{n}}$.