Tangent Lines and Planes

Suppose we have a surface given by $F(x,y,z)=0$ where we can parametrize $x,y,z$ by $t$. Differentiating with respect to $t$ gives:

$$\frac{\partial F}{\partial x}\frac{dx}{dt}+\frac{\partial F}{\partial y}\frac{dy}{dt}+\frac{\partial F}{\partial z}\frac{dz}{dt}=0$$

If we define $\vec{r}=(x,y,z)$ to be the vector valued function that defines the surface, then we can write the above as:

$$\nabla F\cdot \frac{d\vec{r}}{dt}=0$$

Thus, the equation for a plane that is tangent to the surface at some point $P$ is:

$$\nabla F{|}_p\cdot (x-x_0,y-y_0,z-z_0)=0$$