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Determine whether the lines intersect(have a common point) and if so, find the coordinates of that point?

r(t) = < 4t+3, 4t + 2, 6t + 6>, for -infinity < t < infinity
R(s) = < s + 2, 3s - 1, -4s + 10>, for -infinity < t < infinity

And if r and R describe the paths of two particles, do the particles collide?

My teacher did an example in class but this question differs from this one so I don't know how to do it.

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1 answer
  1. let's re-write each line as
    x = 4t+3
    y = 4t+2
    z = 6t+6

    and

    x = s+2
    y = 3s-1
    z = -4s + 10

    so
    4t+3 = s+2 ---> 4t - s = -1
    4t+2 = 3s-1 --> 4t - 3s= -3

    subtract: 2s= 2
    s = 1 , then t=0

    if 2nd line, when s = 1
    x = 3
    y=2
    z = 6

    in 1st line when t=0
    x=3
    y=2
    z=6

    Yes, they do intersect at (3,2,6)

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