Given $A(0,0,2)$ and $B(3,4,1)$ in $Oxyz$. Find the minimum value of $AX+BY$ with $X$ and $Y$ being 2 points lying in $Oxy$, and $XY=1$.
P/s: I have figured out a solution, but I don't think it is the best one possible. The answer is down below.
P/s: If anyone could help me generalize this problem too it'd be great.
Project $A$ and $B$ onto $Oxy$. Call the projections $A'(0,0,0)$ and $B'(3,4,0)$. The reason for this is that $AX$ and $BY$ are essentially lines connecting $A$ and $B$ to $Oxy$, and for the sum of their lengths to be minimum, they must each be of minimum length, i.e., be perpendicular to the plane. Now check whether $\vec{XY}$ is equal to $\vec{A'B'}$ or $\vec{B'A'}$ so that $XY=1$.