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Math Help - shortest distance between two lines

  1. #1
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    shortest distance between two lines

    Show that the shortest distance between two lines l1 and l2 in R3 is achieved along a line that is perpendicular to both l1 and l2.

    Im having trouble with this question, and its frustrating, does it have something to do with finding the normal vector to both lines?

    please help!
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  2. #2
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    Quote Originally Posted by sterps View Post
    Show that the shortest distance between two lines l1 and l2 in R3 is achieved along a line that is perpendicular to both l1 and l2.

    ...
    The line l_1 passes through the point A (It is labeled g in the attachment)

    The line l_2 passes through the point B (It is labeled h in the attachment)

    The equations of the lines are:

    l_1: \vec x = \vec a + r \cdot \vec u . The vector \vec u is the direction vector of l_1 and it is painted in blue.

    l_2: \vec x = \vec b + r \cdot \vec v . The vector \vec v is the direction vector of l_2 and it is painted in red.

    The vector \vec n is the result of the cross product of the 2 direction vectors:

    \vec n = \vec u \times \vec v

    Use the unit vector of n: \overrightarrow{n^0} = \frac{\vec n}{|\vec n|} to calculate the distance d between the 2 lines:

    d = \frac{(\vec b - \vec a) \cdot \vec n}{|\vec n|} = \frac{(\vec b - \vec a) \cdot (\vec u \times \vec v)}{|\vec u \times \vec v|}
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