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Thread: Shortest distance between skew lines

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    Shortest distance between skew lines

    Say I have 2 skew lines and they've given me the equations of the lines as well as shortest distance between the lines. How would I find the points on the 2 lines, e.g. A on line 1 and B on line 2 such that AB is the shortest distance.
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  2. #2
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    Quote Originally Posted by acevipa View Post
    Say I have 2 skew lines and they've given me the equations of the lines as well as shortest distance between the lines. How would I find the points on the 2 lines, e.g. A on line 1 and B on line 2 such that AB is the shortest distance.
    1. Draw a sketch.

    2. A with $\displaystyle \vec a$ lies on $\displaystyle l_1$ that means: $\displaystyle \vec a = \overrightarrow{p_1} + k \cdot \vec u$.

    B with $\displaystyle \vec b$ lies on $\displaystyle l_2$ that means: $\displaystyle \vec b = \overrightarrow{p_2} + t \cdot \vec v$.

    The vector $\displaystyle \overrightarrow{AB}$ is collinear to $\displaystyle (\vec u \times \vec v)$

    3. Solve for (k, t, s):

    $\displaystyle
    \left( \overrightarrow{p_2} + t \cdot \vec v \right) - \left( \overrightarrow{p_1} + k \cdot \vec u \right) = s \cdot (\vec u \times \vec v)$
    Attached Thumbnails Attached Thumbnails Shortest distance between skew lines-mindistwndschfgerade.png  
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