Geometric Interpretation of 3D Minimum Distance Using Lagrange

Question

When we try to maximize/minimize the value of $f(x,y) = 2x + y$ on the unit circle $x^2 + y^2 = 1$, we look at the level curves of $2x + y$ and find the $z$ value at which the line is tangent to the circle.

How does this logic apply to a function of 3 variables? For example, if I want to find the minimum distance from the origin to the intersection of two planes, I would set $f = x^2+y^2+z^2$ and $g_1,g_2$ as the two planes. When we look at the level surfaces of $f$, do we want the expanding sphere to be tangent to both planes? Or do we want the sphere to just touch a point on the intersection of the two lines? Since we want $\nabla{f} = \nabla{g_1} + \nabla{g_2}$, I would think it's the former, but I can't visualize how that allows the sphere to also be on the line of intersection.

Answer 1

The intersection of two planes will be a "straight line" through 3-space. The question is only interesting when this line doesn't go through the origin so this is the case we shall consider.

As you said, the level surfaces of $f$, will be spheres centered at the origin. We are looking for the level surface where the planar intersection (the straight line) is tangent to the sphere. The point of intersection will be the minimizer.