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Vector Planes & Orthogonality - Help!

  1. Apr 25, 2009 #1
    Vector Planes & Orthogonality -- Help!

    I must be doing something really stupid, and I'll kick myself when you point it out, but I'm having difficulty with this question:

    Find the unit normal to the plane x + 2y – 2z = 15. What is the distance of the plane from the origin?

    OK, so I know I need three points in the plane to identify it. I found points:

    A = (15,0,0)
    B = (15,1,1)
    C = (13,2,1)

    They all seem to satisfy the equation, right?

    So now I find two nonparallel vectors in this plane:

    A->B = 0i + 1j + 1k
    A->C = -2i + 2j + 1k

    They do indeed join those points, right?

    So now I want to find an orthogonal vector, so I take the cross product by calculating the determinant of the 3x3 matrix:

    [i j k]
    [0 1 1]
    [-2 2 1]

    = (1-2)i - (-2-0)j + (0+2)k
    = -i + 2j + 2k

    This vector ought to be orthogonal to vectors A->B, A->C and B->C (and their negatives). But I try to test it with the usual method of adding up the parts and seeing if they result in zero, using A->C as my comparison vector:

    (-2 * -1) + (2 * 2) + (1 * 2) = 8

    Why doesn't this equal zero? I have a feeling that the middle (j) term is wrong, because if that was (2 * -2) then the whole thing would equal zero, but I don't see where I've got my signs wrong in the method.

    Any help would be much appreciated.

    Cheers!
     
    Last edited: Apr 25, 2009
  2. jcsd
  3. Apr 25, 2009 #2
    Re: Vector Planes & Orthogonality -- Help!

    The second term in the sum should be -(0 - (-2))j = -2j.
     
  4. Apr 25, 2009 #3
    Re: Vector Planes & Orthogonality -- Help!

    Oh, I see, so it doesn't wrap around! That's probably why it's "+ i - j + k". Thanks for that. I knew it was something simple. :smile:
     
  5. Apr 26, 2009 #4

    HallsofIvy

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    Re: Vector Planes & Orthogonality -- Help!

    Actually, you should have learned early that if a plane is given by Ax+ By+ Cz= D, the <A, B, C> is normal to the plane. You don't need to do all that work. Since the plane is given as x + 2y – 2z = 15, you know that a normal vector is <1, 2, -2>.
     
  6. Apr 26, 2009 #5
    Re: Vector Planes & Orthogonality -- Help!

    Well for some reason that's not in our course, but I shall certainly be using it from now on!
     
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