Find the point on the plane ax + by + cz = d at minimum distance from the origin using the method of lagrange multipliers.

The distance between some point and the origin is given by



so this is the function we're trying to minimize. But notice that and attain their critical points at the same , so we can solve the same problem by minimizing instead.

So let's take the Lagrangian to be



with partial derivatives (set equal to 0)






Now, notice that




and we can use this to solve for . We get





At this critical point, we get a minimum distance of