Algebra

Let L1 be the line passing through the points Q1=(−2, −3, −4) and Q2=(−5, −2, −3) and let L2 be the line passing through the point P1=(−27, 31, −25) with direction vector →d=[−6, 9, −6]T. Determine whether L1 and L2 intersect. If so, find the point of intersection Q.

Having a hard time with this one..

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  1. direction of L1 is [-3, 1, 1]
    equation of L1
    L1 = (-2,-3,-4) + s(-3,1,1)

    equation for L2
    L2 = (−27, 31, −25) + t(−6, 9, −6)

    so : -2 - 3s = -27 - 6t ---> 6t - 3s = -25
    and -3 + s = 31 + 9t ---> s = 9t + 34
    using substitution :
    6t - 3(9t+34) = -25
    6t - 27t - 102 = -25
    -21t = 77
    t = -11/3, then s = 9(-11/3) + 34 = 1

    check in third equation:
    -4 + s = -25 - 6t
    LS = -4 + 1 = -3
    RS = -25 - 6(1) = -31
    LS ≠ RS
    They do not intersect.
    Check my arithmetic, I did not write it down first.
    The method is correct.

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  2. thank you so much!

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