Ok so I'm doing a project on multivariable optimization, and I am using the lagrange method, and I can use it correctly. However, I cannot explain (which i have to do as part of my project) how the lagrange method works..i am going the route of gradient vectors, and i get confused when i have the gradient vector of f(x,y,z) = the gradient vector of g(x,y,z) * lamda
where lamda is the langrange multiplier.
i dont get why the two gradient vectors are parallel, and thus need a multiplier to determine the magnitude. why arent the gradient vectors the same? why are the parallel?