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Determining whether three points lie on a straight line in three dimension

  1. Aug 26, 2008 #1
    1. The problem statement, all variables and given/known data
    Determine whether the points lie on straight line
    A(2, 4, 2) B(3, 7, -2) C(1, 3, 3)

    2. Relevant equations



    3. The attempt at a solution
    I've looked up at the equation for lines in three dimension, and it appears to be
    x=x_0+at
    y=y_0+bt
    z=z_0+ct

    i tried to take the x y z for A and B and try to solve for a, b, c. Then if the same a, b, c work for BC, then ABC is on a line. That is my thought, but i can't manage to do the first part. I don't know how to use the information given and the equations to start with...

    Anyone please help me with this. This is my first time working with 3-dimensional coordinate system...
     
  2. jcsd
  3. Aug 26, 2008 #2

    Dick

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    You don't have to work all that hard to get the equation for the line. In vector form the equation is [x,y,z]=A+(B-A)*t. Do you see why that gives you [x,y,z]=A at t=0 and [x,y,z]=B at t=1?? Can you translate that into equations for x, y and z?
     
  4. Aug 26, 2008 #3
    This part i understand.
    But i'm still not sure about how to translate that in to equations for x, y and z.
     
  5. Aug 26, 2008 #4

    Dick

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    B-A=[1,3,-4], right? So you have [x,y,z]=[2,4,2]+[1,3,-4]*t. I read off x=2+t. I just equated the first component of the two sides. What do you get for y and z?
     
  6. Aug 26, 2008 #5
    i see...

    so y= 4+3t and z=2-4t?
    and from here, i can use the x, y, z equation for points BC to see if it's a line?
     
  7. Aug 26, 2008 #6

    Dick

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    Nah, just see if C is on the line, you don't need another set of equations. If there is a t that solves all three, then it's on the line. If not, not.
     
  8. Aug 27, 2008 #7
    Thank you very much. I've got it
     
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