Find the points that give the shortest distance between the line $(2,3,1)+s(1,2,-1)$ and $(1,2,0)+t(2,-3,5)$ using lagrange multipliers
The problem is that these lines aren't giving in the form like $x+y=2$, or something like $x^2+y^2=4$
They aren't defined "via an equation", and so I am unsure how to do this problem.
All the questions on MSE have lagrange multipliers of equations, something like Find points that give the shortest distance between $y = x^2$ and $y-x+2=0$ using Lagrange multipliers
So I don't even know where to start here?
We know the distance formula squared is like $(x_\text{on line 1}-x_\text{on line 2})+(y_\text{on line 1}-y_\text{on line 2})+(z_\text{on line 1}-z_\text{on line 2})$
$d^2 = ((2+s) - (1+2t))^2 + ((3+2s) - (2-3t))^2 + ((1-s) - (5t))^2\\ d^2 = (1+s-2t)^2 + (1+2s+3t)^2 + (1-s - 5t)^2\\ d^2 = 38t^2 + 6s^2 + 3 -8t +4s+ 18 st$
$s, t$ are unconstrained, so there is no further constraint to apply.
Take the partials with respect to $s, t$ and set them equal to $0.$
$76 t + 18 s - 8 = 0\\ 12 s + 18 t + 4 = 0$
and that is a system of linear equations.