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Basis for Plane Perpendicular to a Line

  1. Feb 10, 2010 #1
    1. The problem statement, all variables and given/known data
    Let L be the line in R^3 spanned by v1=(1,1,1)

    Find a basis (v2,v3) for the plane perpendicular to L, and verify that B=(v1,v2,v3) is a basis for R^3.

    2. Relevant equations

    3. The attempt at a solution
    I know that if two vectors are perpendicular or orthogonal that their dot product is equal to zero. However, I am not sure how to find a plane that is perpendicular to a vector.
  2. jcsd
  3. Feb 10, 2010 #2


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    Hint: If you cross v1 with anything not parallel to it you will get a vector v2 perpendicular to v1. What direction will v2 cross v1 have?
  4. Feb 11, 2010 #3


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    You could also solve the equation

    [tex]\begin{pmatrix}1 & 1 & 1\end{pmatrix}\begin{pmatrix}x \\ y \\ z\end{pmatrix} = 0[/tex]

    You'll get a two-dimensional solution from which you can read off two basis vectors.
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