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There are two lines in $ \Bbb R^3 $ given in parametric form: $$ l_1: \left\{ \begin{aligned} x &= x_1 +a_1t\\ y &= y_1 +b_1t \\ z &= z_1 +c_1t \\ \end{aligned} \right. $$ $$ l_2: \left\{ \begin{aligned} x &= x_2 +a_2s \\ y &= y_2 +b_2s \\ z &= z_2 +c_2s \\ \end{aligned} \right. $$

What's the simplest method (or formula) for finding (the shortest) distance beteen them?

example I'm doing:

$ l_1: \left\{ \begin{aligned} x &= 0 \\ y &= -1 - 2t \\ z &= -2t \\ \end{aligned} \right. $

$ l_2: \left\{ \begin{aligned} x &= 3s \\ y &= 1 - s \\ z &= 2 + 4s \\ \end{aligned} \right. $

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Hint: Find a vector $\mathbf A$ perpendicular to both lines and then find the projection onto $\mathbf A$ of any vector joining $\ell_1$ and $\ell_2$. Pictures help.

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  • $\begingroup$ That's a very clever and concise way to go about it, giving a simple form for the answer. $\endgroup$
    – Eric Auld
    Commented Jun 25, 2013 at 13:03

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