A problem with inner products on Hilbert space

Given $f \in \mathcal H$ and $u$, $v \in \mathcal H\!\setminus\!\{0\}$, find the most general constraints on $u$ and $v$ to guarantee $$\langle u, f\rangle v = \langle v, f\rangle u \qquad \forall \, f \in \mathcal H$$ Here $\langle \cdot\,,\cdot \rangle : \mathcal H \!\times\! \mathcal H \to \mathbb C $ denotes the hermitian inner product.

Clearly one can just guess $u = av$, with $a \in \mathbb{R}\!\setminus\!\{0\}$, and indeed it works, but is it the most general solution? Is there a much rigorous strategy apart from guessing?

Choosing a fixed $f$ such that $ \langle u, f \rangle \neq 0$ we see that $v=au$ where $a=\frac {\langle v, f \rangle } {\langle u, f \rangle }$. (Note that $f$ exists: we can take $f=u$). $a=0$ is not possible because that would make $v=au=0$.