I am aware of how to find an equation of the tangent place to a surface that is given as the graph of a function $z = g(x,y)$. Here one finds a normal vector by essentially taking the partial derivatives.
My question is if there is a formula that can be used when the surface is given by a general parametrization $\vec{r}(u,v)$. I would assume that there are still some partial derivatives and maybe a cross product somewhere, but I am not quite seeing it.
(I am just asking out of curiosity.)
You know that your plane is parallel to $\vec r_u = \partial\vec r/\partial u$ and $\vec r_v = \partial\vec r/\partial v$, and also passes through point $\vec r(u,v)$. Can you write down the equation from those hints?