To see the next important property we should take a look at the set of points $\mathbf r$ for which $\mathbf r\cdot \mathbf a = C$ for some fixed constant $C$.
You can see that this set of points is exactly the set of points for which the projection onto $\mathbf a$ is the point $C \cdot \dfrac{\mathbf a}{|\mathbf a|}$ and they form a hyperplane orthogonal to $\mathbf a$.

I guess for $\mathbf r\cdot \mathbf a = C$ the point of projection is $\mathbf r \cos \theta$ which is of magnitude $C/|\mathbf a|$ and has direction of $\hat{a}$ (unit vec along ${\mathbf a}$) so the point is $C \cdot \dfrac{ \hat{\mathbf{a}}}{|\mathbf a|}$ which evaluates to the point $C \cdot \dfrac{\mathbf a}{|\mathbf a|^2}$