Suppose we have a plane in $\mathbb{R}^3$ whose equation is given by $$z=ax+by,$$
and let $\textbf{n}(-a,-b,1)$ be a vector normal to this plane.
How can we intuitively explain why the projection of $\textbf{n}$ onto the $xy$-plane, $\textbf{n}'(-a,-b)$, points toward the steepest slope (with respect to a point in the $xy$-plane)?
