Prove $S\cap S ^\bot=\{0\}$

Question

Let $S$ be a nonempty subset of the inner product space $V$ and $S^\bot$ be the orthogonal complement of $S$.

Prove $S\cap S ^\bot=\{0\}$

Answer 1

If $s\in S$ is also in $S^{\perp}$ then by definition $\langle s, r\rangle=0$ for all $r\in S.$ Then taking $r=s$ gives $s=0.$

Answer 2

Hint $\langle x, x\rangle >0$ if $x\ne 0$.