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Let L1 be the line passing through the points Q1=(4, −1, −2) and Q2=(1, 0, −1) and let L2 be the line passing through the point P1=(−6, 21, −8) with direction vector →d=[−3, 9, −3]T. Determine whether L1 and L2 intersect. If so, find the point of intersection Q.

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  1. for L1, direction vector is (3,-1,-1)
    and an equation is
    x = 1 + 3t
    y = -t
    z = -1 - t

    for L2, a possible equation is
    x = -6 - 3k
    y = 21 + 9k
    z = -8 - 3k

    if they intersect,
    1+3t = -6-3k AND -t = 21+9k AND -1-t = -8-3k

    using the first two equations,
    1 + 3(-21-9k) = -6-3k
    -62 - 27k = -6-3k
    -24k = 56
    k = 56/-24 = -7/3
    then t = -21 - 9(-7/3) = 0

    does this satisfy the third equation?
    LS = -1-0 = -1
    RS = -8-3(-7/3) = -1
    yes it does!

    so they do intersect and the point of intersection is

    x = 1+3(0) = 1
    y = 0
    z = -1-0 = -1

    they intersect at (1,0,-1)

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