Say there was a function $f(x, y, z)$ constrained by $g(x, y, z) = c$.
I understand intuitively/spatially/visually why at the max or min, $\nabla f$ and $\nabla g$ are parallel. (I saw a very good explanation which used contour lines.)
But I'm having trouble understanding the case with multiple constraints, like: $f(x, y, z)$ constrained by $g(x, y, z) = c$ and $h(x, y, z) = k$.
Why is $\nabla f$ in the plane spanned by $\nabla g$ and $\nabla h$? ($\nabla f$ = $\lambda \nabla g$ + $\mu \nabla h$)
Aren't gradients of the same dimensions always in the same plane already? (For some 3D surface, the gradient $\nabla f = (\frac{\partial f}{\partial x} , \frac{\partial f}{\partial y})$ is always in the x-y plane.) So what new information does this give us?
I do understand it when it's written out like $L(x, y, z, \lambda, \mu) = f + \lambda (g - c) + \mu (h - k)$, since when you set $\nabla L$ = 0, it will give the above equation. But I would prefer to have an intuitive/spatial/visual grasp as well.