Proving line is perpendicular using partial derivatives

Question

Consider a smooth surface given by the function $z = f(x,y)$, such that the partial derivatives of $f(x,y)$ exist. Suppose $Q$ is a point that does not lie on the surface, and $P$ is the nearest point on the surface to $Q$. Show that the line through $P$ and $Q$ is perpendicular to the surface at $P$.

Answer 1

At the point $P(x_p,y_p,z_p)$; the tangent plane to the surface $z=f(x,y)$ has following cartesian equation :

$$z-z_p=\frac{\partial f}{\partial x}(x_p,y_p)(x-x_p)+\frac{\partial f}{\partial y}(x_p,y_p)(y-y_p)$$

thus the normal vector is $$\vec{PQ}=\lambda (\frac{\partial f}{\partial x}(x_p,y_p),\frac{\partial f}{\partial y}(x_p,y_p),-1)$$