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

  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

    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. Dick

    Dick 25,913
    Science Advisor
    Homework Helper

    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. This part i understand.
    But i'm still not sure about how to translate that in to equations for x, y and z.
  5. Dick

    Dick 25,913
    Science Advisor
    Homework Helper

    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. 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. Dick

    Dick 25,913
    Science Advisor
    Homework Helper

    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. Thank you very much. I've got it
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