$V$ is an inner product space and $u, v$ belong to $V$. I need to prove that $\langle u,v\rangle = 0$ iff $\|u\| \le \|u+cv\|$ for all scalars $c$.
So far to show the backward proof, I've established that $0 \le \overline{c}\langle u,v\rangle + c\langle v,u \rangle + c\langle v,v\rangle $ but don't think I am on the right track. Any hints or methods would be much appreciated!