Show that $\oint (\vec{c} \cdot \vec{r} )d\vec{l} = \vec{a} \times \vec{c}$

Question

I need to prove part c). This is what I've tried (using Stokes Theorem):

$$\nabla \times \vec{c}T = T(\nabla \times \vec{c}) - \vec{c} \times (\nabla T)$$ $$=(\vec{c} \cdot \vec{r})(\nabla \times \vec{c})- \vec{c} \times (\nabla (\vec{c} \cdot \vec{r}))$$

But I'm stuck. Any tips?