Find the point on the plane ax + by + cz = d at minimum distance from the origin using the method of lagrange multipliers.
Accepted Solution
A:
The distance between some point [tex](x,y,z)[/tex] and the origin is given by
[tex]f(x,y,z)=\sqrt{x^2+y^2+z^2}[/tex]
so this is the function we're trying to minimize. But notice that [tex]f(x,y,z)[/tex] and [tex]f(x,y,z)^2[/tex] attain their critical points at the same [tex](x,y,z)[/tex], so we can solve the same problem by minimizing [tex]x^2+y^2+z^2[/tex] instead.