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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.

user518704
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    Welcome to stackexchange. You are more likely to get answers rather than downvotes and votes to close if you edit the question to show us what you tried and where you are stuck. (This is your second question here. You showed no work on the other one either.) – Ethan Bolker Mar 14 '18 at 17:10
  • How will I multiply $X$ and $Y$ to $A$ and $B$, respectively? Is $BY=3Y+4Y+1Y$? – John Glenn Mar 14 '18 at 17:54
  • $A$, $B$, $X$, $Y$ are points, not vectors. – user518704 Mar 14 '18 at 23:12
  • I got $X=(\frac{8}{5},\frac{32}{15},0)$, $Y=(\frac{11}{5},\frac{44}{15},0)$. $AX+XY+YB=6$ – CY Aries Mar 15 '18 at 16:16

1 Answers1

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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 $XY=1$.

user518704
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