Where does lambda come from in Lagrange multipliers? Specifically related to find maximum and minimum values on a constraint.

Question

The book states that the $\nabla f(x,y,z) = \lambda\nabla g(x,y,z)$ It talks about the slope of the tangents being parallel, but wouldn't they technically just be same line? It also brings up norms in the definition. I don't understand how that applies. Could someone simplify a definition for me that hits the major points? Over 6 views and no answers maybe I am too moronic to be helped.

Answer 1

At a stationary value of the objective $f(x,y,z)$, subject to constraint $g(x,y,z)=0$, the level surface of the objective is kissing the constraint surface. This means the normals to the two surfaces are parallel which is expressed by there existing a real scalar $\lambda$ such that $\nabla f=\lambda \nabla g$.

The gradient $\nabla h(\bf{x})$ is a vector pointing in the direction that $h(\bf{x})$ is increasing most rapidly at $\bf{x}$, aka in the direction of the normal to the level surface of $h$ through $\bf{x}$.