# Shortest distance between two lines and find points on each line [duplicate]

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.

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