How to prove that $\oint_{C} q[\vec{v}\times \vec{B}] \cdot d\vec{r}=0$

Question

How to prove that $\oint_C q[\vec{v}\times \vec{B}] \cdot d\vec{r}=0$ rigorously, where $q[\vec{v}\times \vec{B}]$ is the magnetic force acting on a positive change q, $\vec{v}$ is the velocity, and $\vec{B}$ is the magnetic field. I can prove it qualitatively by saying that $\vec{v}$ is is in the same direction as d$\vec{r}$, and since $\vec{v}\times \vec{B}$ is normal to $\vec{v}$, therefore to d$\vec{r}$, then $q[\vec{v}\times \vec{B}] \cdot d\vec{r}=0$ and consequently $\oint_C q[\vec{v}\times \vec{B}] \cdot d\vec{r}=0$, but this doesn't look like a rigorous proof to me, so how can I prove it rigorously?