Eqn of Plane passing through a point and perpendicular to another plane's trace

  1. 1. The problem statement, all variables and given/known data
    Here's the problem:Find the equation of the plane passing through the point (-3,1,4) and perpendicular to the trace of the plane x-3y+7z-3=0 in the xy plane.

    2. Relevant equations

    to me this should be as easy as finding the two coordiates of the plane's traces on the x and y axis. ie (3,0,0) and (0,-1, 0). Since these two points and the point (-3,1,4) determine the plane.

    3. The attempt at a solution
    since the general form of the plane is Ax+By+Cz+D=0. I should get 3 equations that look like this:
    1) 3A+D=0
    2) -B+D=0
    3) -3A+B+4C+D=0
    1) implies that -D/A=3 and thus D=-3 and A=1
    substituting into eqn2 yields B=-3
    and sunstituing into eqn3 gives C=9/4
    so the eqn should be x-3y+9/4Z-3=0 or 4x-12y+9z-3=0
    However the answer in the book is 3x+y+8=0. I find that the book is rarely wrong and usually it's me that is confused.

    Please advise on where I went wrong
    Thanks so much
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,310
    Staff Emeritus
    Science Advisor

    These 3 points give the plane containing the trace in the xy plane, not perpendicular to it. The trace of the plane in the xy-plane is given, of course, by x- 3y- 3= 0 and z= 0. If you use y itself as parameter, x= 3y+ 2, y= y, z= 0 are parametric equation of that line. A vector pointing in the direction of that line is <3, 1, 0>. You want the equation of a plane containing (-3, 1, 4) and having normal vector <3, 1, 0>.

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