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Math Help - Shortest Distance between 2 straight lines

  1. #1
    Newbie
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    Exclamation Shortest Distance between 2 straight lines

    Consider two non-parallel straight lines in 3-dimensional space. The fi rst line can be
    described, in Cartesian coordinates (x; y; z), by the parametric equations
    x(u) = x1 + ua1 ; y(u) = y1 + ub1 ; z(u) = z1 + uc1
    for some set of numbers (x1; y1; z1) and (a1; b1; c1). Likewise the second line can described
    by x(v) = x2 + va2 ; y(v) = y2 + vb2 ; z(v) = z2 + vc2
    for another set of numbers (x2; y2; z2) and (a2; b2; c2).

    (a) Verify that the shortest distance D between this pair of lines is given by
    D =

    [x1 􀀀 x2 y1 􀀀 y2 z1 􀀀 z2]
    [a1 b1 c1]
    [a2 b2 c2]
    /
    s quare root
    ([a1 b1]
    [a2 b2])^2

    +

    ([b1 c1]
    [b2 c2])^2


    +

    ([c1 a1]
    [c2 a2])^2


    Be sure to point out where in your analysis you use the assumption that the lines are not parallel.
    (b) Now suppose that the two lines are parallel. Find a new formula for D.
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  2. #2
    MHF Contributor

    Joined
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    Re: Shortest Distance between 2 straight lines

    Don't you think it would be a good idea to show us what you have done on this problem?

    Also, your notation is almost impossible to understand. Are those arrays supposed to be matrices?
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