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Homework Help: Finding whether two lines intersect each other in 3dimensional space

  1. Jan 30, 2010 #1
    1. The problem statement, all variables and given/known data
    do the lines (x,y,z) = (5+2t, 3+2t,1-t) and (x,y,z) = (13-3r, 13-4r, 4-2r) intersect? If so, at what point? If not, how do we know?

    2. Relevant equations

    3. The attempt at a solution

    I just do not know where to begin...
    I mean what do you do with the variables t and r.
    Do those values constanly change?
    If you want to direct me to some useful information to help me understand this concept.. feel free to do so
  2. jcsd
  3. Jan 30, 2010 #2
    If they do intersect, then the x component of the first line must equal the x component of the second line, and similar for the y and z components.

    So, perhaps you should set the equations equal to each other...
  4. Jan 30, 2010 #3
    does this mean
    1-t=4-2r ?
    oh i guess when t=1 and r=2, those two lines intersect each other
  5. Jan 31, 2010 #4


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    Science Advisor

    Yes, it happens that when t= 1 and t= 2, all three equations are satisfied. And you can do more. By putting t= 1 into the equations for that line or by putting r= 2 into the equations for the second line, you get x= 5+2= 7, y= 3+ 2= 5, and z= 1-t= 0 or, equivalently, 13- 6= 7, y= 13- 8= 5, 4- 4= 0 so the two lines intersect at (7, 5, 0).

    Of course, in three dimensions, "most" lines do NOT intersect. You could always solve two of the equations, say, the x and y equations, for s and t. But then you would have to check in the z equation to see if that also was satisfied.
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