Given a vector $u$ such that $\langle v,u\rangle=0$ for every vector $v$, prove that $u=0$

Question

Suppose $\langle \cdot, \cdot\rangle $ is an inner product on $\mathbb{F}^n$. Given a vector $u$ such that $\langle v,u\rangle=0$ for every vector $v$, prove that $u=0$.

Here $\mathbb F$ is a field. Trying to figure out how to approach this, although I am sure I am missing something simple. Any help would be appreciated.

Answer 1

What happens if $u=v$?

Since $\langle u,u\rangle=0$, the vector $u$ is zero, because $\langle\cdot,\cdot\rangle$ is an inner product.