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I'm given the following two lines:

$L_1$: $P_1=(−13, 3, 14)$ with direction vector $d_1=(2, −1, −2)$

$L_2$: $P_2=(5, 4, 4)$ with direction vector $d_2=(−2, 1, 0)$

I'm then asked to find the shortest distance $d$ between these two lines, and then find a point, $Q_1$, on $L_1$, and a point, $Q_2$, on $L_2$ so that $d(Q_1,Q_2) = d$.

So far, I've determined that the shortest distance between these two points can be solved with a projection of the vector $\vec{P_1P_2}$ onto the direction vector found by the cross product of $d_1$ and $d_2$. In this case, it is $(4,8,0)$, or a magnitude of $4\sqrt{5}$.

I'm not really sure how to determine the two points now that I've gotten the distance, any tips or explanations would be appreciated.

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  • $\begingroup$ You can find many questions in the handy list of related questions at right that show you how to do this. $\endgroup$
    – amd
    Sep 26 '18 at 2:42
  • $\begingroup$ @amd My searching skills have proven to not be what they once used to... Thanks! $\endgroup$
    – simplyme
    Sep 26 '18 at 3:11
  • $\begingroup$ No worries. For some reason the search function in the UI doesn’t work nearly as well as the one that finds related questions. $\endgroup$
    – amd
    Sep 26 '18 at 6:57
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HINT

Let consider the plane orthogonal to a line and containing the other one.

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Solve the subproblem ("what is the distance from a point to a line?" (you should have this in your notes or book)) then perform a constrained optimization problem by allowing that point to vary over the line.

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If you parameterize both lines, then the square distance between two points, one on each line, is quadratic in the parameters. Do you know how to minimize a function of two variables?

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