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Orthonormal Basis of a Plane

  1. May 24, 2006 #1
    I need to find the Orthonormal Basis of this plane:

    x - 4y -z = 0

    I know the result will be the span of two vectors but I'm not sure where to start. Any hints?


  2. jcsd
  3. May 24, 2006 #2


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    First find a basis by finding two independent vectors that satisfy that equation. This is easy: find one non-zero vector satisfying that equation with z-component 0, and find another satisfying that equaiton with y-componenet 0. Next, orthogonalize this basis using Gramm-Schmidt. Finally, normalize it by dividing the two orthogonal vectors you have by their own norms.
  4. May 24, 2006 #3
    So set (y=1, z=0) and (y=0, z=1)

    Get two vectors:

    (4,1,0) and (1,0,1)


    (4/sqrt(17), 1/sqrt(17), 0) and (1/sqrt(2), 0, 1/sqrt(2))
  5. May 24, 2006 #4


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    To satisfy the "ortho" part of Orthonormal you need to verify that the dot product of your 2 vectors is 0.
  6. May 24, 2006 #5
    Ah thanks,

    so e1= (1/sqrt(2), 0, 1/sqrt(2))

    e2 = (2/3, 1/3, -2/3)
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