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Interseccion of two planes in R3

  1. Oct 11, 2012 #1
    How do you find the intersection of two planes in R3? The direction vector would be the cross product between the two normal vectors I imagine. So, how do I go about finding a point that lies in both planes so I can find the equation of the line?

    Thanks :)
  2. jcsd
  3. Oct 11, 2012 #2
    What do you know about the equation of a plane in R3?
  4. Oct 11, 2012 #3
    A plane is defined by a normal vector and a point. It can be written as
    ax+by+cz=d where (a,b,c) is the normal vector and d is <(x1,y1,z1),(a,b,c)>
  5. Oct 11, 2012 #4
    Exactly so.

    The general linear form in 3D is a plane, not a line.

    In fact there is no single "equation of a line in 3D", which is probably why you can't find one.

    Two planes intersect in a line so a line is defined by two planes.

    A line has to be defined by two equations, not one.

    a1x+b1y+c1z=d1 = P1
    a2x+b2y+c2z=d2 = P2

    For an alternative pair of equations see here


    However any linear combination of P1 and P2 will also contain this line. This can be described by the parameter λ such that

    P1 + λ(P2) = 0

    This is referred to as a pencil of planes or a fan of planes or a sheaf of planes.

    Wolfram have a good picture.
    Last edited: Oct 12, 2012
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