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Math Help - Determining whether the lines are parallel, intersect, or skew

  1. #1
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    Determining whether the lines are parallel, intersect, or skew

    I have a problem where i don't have the solution to since i like to practice odd problems.
    I am not sure if i have done this correctly.

    Determine whether the lines

    L1: 1 + t, y = 2 + 3t, z = 3 + t

    L2: 1 + t, y =  3 + 4t, z = 4 + 2t

    are parallel, intersecting, or skew. If they intersect find the point of intersection.

    Attempt: Take L2 and re-write it using t=s to make it simpler
    L2: 1 + s, y =  3 + 4s, z = 4 + 2s

    Make them parametric:

    (1) 1+t = 1+ s
    (2) 2+3t = 3+4s
    (3) 3+t = 4+2s

    Solve (1) for t

    1+t = 1+ s
    t = s

    Solve(2) for s
    2+3t = 3+4t
    s = -1

    so, t = -1 and s = -1

    Solve (3) by subbing in t and s

    3+(-1) = 4+2(-1)
    2 = 2

    Therefore, two lines intersect. Not parallel.

    To find the point of intersection:

    Solve L1 for t = -1

    L1: 1 + (-1), y = 2 + 3(-1), z = 3 + (-1)

    point: (0, -1, 2)

    Is this correct? Or did i do this completely wrong?
    Thank you
    Last edited by icelated; May 1st 2013 at 04:36 PM.
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  2. #2
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    Re: Determining whether the lines are parallel, intersect, or skew

    Hey icelated.

    Confirming you answers with Octave, we get:

    >> >> A = [1, -1, 0; 4, -3, -1; 2, -1, -1]
    A =

    1 -1 0
    4 -3 -1
    2 -1 -1

    >> rref(A)
    ans =

    1 0 -1
    0 1 -1
    0 0 0

    This confirms your results as correct.
    Thanks from icelated
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  3. #3
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    Re: Determining whether the lines are parallel, intersect, or skew

    Thank you so much for responding.
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