Differentiable submanifold

Hello.

Let $\displaystyle M \subset \mathbb R^n$ be a differentiable submanifold (with the induced metrics from $\displaystyle \mathbb R^n$), $\displaystyle q \in \mathbb R^n\setminus M$,

and $\displaystyle g:M\to \mathbb R, \quad x \mapsto \left \| q-x \right \|$ has a minimum $\displaystyle p \in M$.

Proof, that the vector $\displaystyle q-p$ is orthogonal to $\displaystyle T_p M$.

$\displaystyle T_p M$ is the tangent space at $\displaystyle p \in M$.

Thanks in advance for help!

Buy,

- Alexander -