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Vector geometry - Intersection of lines

  1. Mar 2, 2012 #1
    1. The problem statement, all variables and given/known data
    I have 2 parametric vector equations (of a line)

    r(t) = (2,-4,4) + t(1,-3,4)
    s(t) = (1,-1,0) + t(2,-1,1)

    how do i find the coordinates for which they intersect each other?
    The answers is (1,-1,0)

    2. Relevant equations

    x=a+λv, for some λ in ℝ (parametric vector form of line)

    3. The attempt at a solution
    As in high school, with the form y=mx+b i would make the 2 equations equal to each other, solve for x, then substitute back into either equations to find y.

    I've tried making the (x,y,z) components equal to each other, solve for 't' and substitute back in but i can't get the answer in the back of the book

    parametic equations
    for r(t): x = 2+t, y= -4-3t, z= 4+4t
    for s(t): x = 1+2t, y= -1-t, z= t

    Now i did
    2+t = 1+2t
    t = 1

    substituting back into x=2+t: x = 3

    i did this for also y and x components and got (3, 1/2, -4/3)
    I have a feeling that this method isnt correct :s
  2. jcsd
  3. Mar 2, 2012 #2


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    Change the notation to
    r(t) = (2,-4,4) + t(1,-3,4)
    s(u) = (1,-1,0) + u(2,-1,1)
    the parameters t and u don't have to be the same for the lines to intersect each other.
  4. Mar 2, 2012 #3
    alright. Now how do i solve it?
  5. Mar 2, 2012 #4


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    Homework Helper

    The same way you tried before. Equate components of the vectors and solve them. Try it. The first component gives you 2+t=1+2u. Solve that for t and substitute into the rest.
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