Let $S$ be a surface in $\mathbb{R}^3$ which is locally defined by a level set of some smooth function. Let $M$ be a point which is not on the surface. First of all, I would like to show that there always exists a unique point $N$ which is on the surface and that minimizes the distance $MN$. Secondly I would like to show that $\overrightarrow{NM}=d\cdot\vec\nu(N)$. I think this is a standard theorem of surface theory but I am not able to prove the following points. Could somebody provide some ideas or references?
Many thanks!