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Orthogonal complement of the intersection of 2 planes

  1. May 2, 2015 #1
    1. The problem statement, all variables and given/known data
    Let W be the intersection of the two planes: x-y+z=0 and x+y+z=0
    Find a basis for and the dimension of the orthogonal complement, W

    2. Relevant equations


    3. The attempt at a solution
    The line x+z=0 intersects the plane, which is parameterized as t(1, 0, -1)
    Then W is the plane x-z=0
    Then the nullspace of this plane is (1, 0, 1) which is the basis for W
    And the dimension is 1?
    Am I even in the right ballpark here? :D
     
  2. jcsd
  3. May 2, 2015 #2

    haruspex

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    You need a constraint on y as well to make it a line (but you got that right in the parametric form).
    Correct.
    The only context I know for the term null space is in connection with transformations, and there is no transformation being discussed here.
    The vector you state is a basis for W.
     
  4. May 2, 2015 #3
    Okay sweet!
    So if (1,0,1) is the basis for W, shouldn't there be one more basis vector since W is a plane and a plane is 2 dimensional?
     
  5. May 2, 2015 #4

    haruspex

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    No, I wrote that it is a basis for W, not W.
     
  6. May 2, 2015 #5
    Okay so how do you find the basis for W then?
     
  7. May 2, 2015 #6

    haruspex

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    It's not 'the' basis, it's 'a' basis.
    You correctly stated the constraint for it, x=z. You just need two independent vectors in it.
     
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