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Calculus

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Explain fully please and thank you! =)

Determine the point of intersection of the lines L1: r= (1,3,7) + s(1,2,-2),seR and L2: v=(0,2,-3) +t(1,1,4), teR.

a) P(1,3,1)
b P(1,3,7)
c) The lines coincide
d) no point of intersection

  • Calculus -

    if they intersect:
    s+1 = t+0 ----> t = s+1
    and
    2s+3 = t+2 ----> t = 2s+1
    and
    -2s+7 = 4t-3

    using the first two equations
    2s+1 = s+1
    s = 0 , then t=1

    test in 3rd equation ...
    LS = -2s+7 = 0 + 7 = 7
    RS = 4t-3 = 4-3 = 1 ≠ LS

    so they cannot intersect
    Do they coincide?
    If they did, they should have the same direction vectors ...
    the first has direction (1,2,-2) and the second has direction ( 1,1,4) , so they are not parallel

    conclusion: We are looking at two lines which are not parallel and which do not intersect

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