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Math Help - Endpoints of shortest distance between Skew lines

  1. #1
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    Endpoints of shortest distance between Skew lines

    I know the forumla to find the distance between two skew lines in space. But, I need to find out the coordinates on the two lines that represent the shortest distance.
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  2. #2
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    Quote Originally Posted by jsmith90210 View Post
    I know the forumla to find the distance between two skew lines in space. But, I need to find out the coordinates on the two lines that represent the shortest distance.
    Let

    L_1:\vec r = \vec a+s\cdot \vec u
    and
    L_2:\vec r = \vec b + t\cdot \vec v

    denote the equations of the two lines. I assume that L_1\cap L_2=\emptyset and that exist \vec n = \vec u \times \vec v

    To calculate the endpoints of the shortest distance between arbitrary points of the two lines the distance vector must be a multiple of the normal vector:

    (\vec a - \vec b)+s\cdot \vec u -  t\cdot \vec v = k\cdot \vec n

    You'll get a system of 3 simultaneous equations. Solve for (s, t, k). Plug in the values of s and t into the appropriate equations to get the endpoints of the distance.
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  3. #3
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    I'm sorry, but I'm not sure i understand the form you're using for the equation of a line. I've learned that the equation for a line is like the following:

    (2+3d)i + (3+2d)j + (1+4d)k

    Where: (2, 3, 1) is a point on the line. And 3i + 2j + 4k would be the vector representing the orientation of the line
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