Constant scalar field on a surface

Question

Suppose that a scalar field $\psi$ is constant on a surface $S$. Let $\mathbf{r}(u,v)$ be a point on the surface $S$. Show that $\partial_{u}\mathbf{r}\times\partial_{v}\mathbf{r}=\lambda\nabla\psi$ where $\lambda$ is a scalar field.

I am not sure how to begin. I would be glad for any hint how to proceed from this.

Answer 1

HINT:

if $\psi$ is constant on $S$ then its gradient is normal to the surface, i.e. it is perpendicular to the tangent plane at $\mathbf{r}$.