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Orthagonal Vectors in 4 Space

  1. Nov 14, 2006 #1
    I have this question that says:
    Find two vectors of norm 1 that are orthagonal to the three vectors u = (2, 1, -4, 0), v = (-1, -1, 2, 2), and w = (3, 2, 5, 4).

    I've tried setting up a system of equations to solve.
    2a + b - 4c = 0
    -a - b + 2c + 2d = 0
    3a + 2b + 4c + 4d = 0

    But when I did that I was left with a free variable. So basically I was wondering if there's another way to do it such as taking the determinate like how you do in 3 space. Except in 4 space.
    Eg.
    i j k
    0 1 0
    1 2 5

    Shane
     
  2. jcsd
  3. Nov 14, 2006 #2

    StatusX

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    Homework Helper

    There will be a whole line of vectors perpendicular to those vectors. But only 2 will have norm 1.
     
  4. Nov 15, 2006 #3

    NateTG

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    Science Advisor
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    If you know how to calculate the determinat of an nxn matrix there is an n-dimensional analog of the cross product:
    [tex]
    \vec{v}=\left| \begin{array}{c c c c}
    \hat{i} & \hat{j} & \hat{k} & \hat{l} \\
    2 & 1 & -4 & 0 \\
    -1 & -1 & 2 & 2 \\
    3 & 2 & 5 & 4 \end{array} \right | [/tex]

    Which will give you a vector perpendicular to the n-1 you already have.
     
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