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Equation of a plane given point and line in parametric form

  1. Sep 1, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the equation of the plane that contains P=(-1,0,1) and r(t)=<3t,t,8>



    2. Relevant equations



    3. The attempt at a solution

    n * <r-r0>=0
    n * <t+2, 2t, 3t> = 0

    I distributed the n, adding the terms and obtained:

    1/t = -2n(n+2n+3n)

    Clearly, I've done something wrong. If someone could point me in the right direction with even how to start this problem correctly, that would be great.

    Thanks!
     
  2. jcsd
  3. Sep 1, 2010 #2
    Ok, I set t=1, found a point P0 (2,2,2) and found two vectors OP and OP0 and took the cross product.
    I then took the dot product of the normal vector I found above and the given point P.

    So, my equation of the plane is 2x + 2x = -1

    Can someone tell me if this is valid/and/or correct?
     
  4. Sep 1, 2010 #3

    Mark44

    Staff: Mentor

    You won't be able to get a unique plane if all you know is a point in the plane and a vector that lies in it.
    I'm not following what you're doing above. I get that n is a normal to the plane, but where did <t + 2, 2t, 3t> come from?
    ???
     
  5. Sep 2, 2010 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    2x+ 2x= -1?? The plane x= -1/4 is the plane parallel to the yz-plane at x= -1/4.
    Did you mean 2x+ 2y= -1 or perhaps 2x+ 2z= -1?

    musicmar, a line and a point not on the line determine a plane. But a "vector" is not a "line". It gives the direction but not specific points. Just given a "vector" we might have a line in the direction of the vector through the given point- and that will not determine a plane.

    Anyway, to check if 2x + 2x= 4x = -1 is a solution, see if it meets the conditions. Does the given point (-1, 0, 1) lie in it? No, it doesn't; [itex]4(-1)\ne -1[/itex]. Nor does it lie in 2x+ 2y= -1 or 2x+ 2z= -1. [itex]2(-1)+ 2(0)= -2\ne -1[/itex] and [itex]2(-1)+ 2(1)= 0\ne -1[/itex].
     
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