A problem asks me to use the method of Lagrange multipliers to find the shortest distance between the straight lines $x=y = z$ and $x = -y, z=2$
(It also warns me that using this method is a bit overkill for this task, so no worries)
I'm actually having trouble to even formulate the problem as the function to minimize, which I assume would be the distance between these two lines, and the constraint, which I'm not entirely sure what would be. Any help is appreciated.
My current thought is that if the two lines are expressed as:
$$\vec{u} = t\pmatrix{1\\1\\1}$$
$$\vec{v} = s\pmatrix{1\\-1\\0} + \pmatrix{0\\0\\2}$$
Then $f(\vec{s},\vec{u})$ should minimize
$$f(\vec{u},\vec{v}) = (t_1 - s_1)^2 + (t_2 + s_2)^2 + (t_3-2)^2$$
Given a certain constraint $g$