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Vectors in R^4 orthogonal to two vectors

  1. Apr 13, 2013 #1
    1. The problem statement, all variables and given/known data

    Find all vectors in $\mathbb R^4$ that are orthogonal to the two vectors
    $u_1=(1,2,1,3)$ and $u_2=(2,5,1,4)$.


    2. Relevant equations

    Gauss-elimination. Maybe cross-product or Gram Schmidt.

    3. The attempt at a solution

    a) Denote a vector $u_3=(v_1,v_2,v_3,v_4)$ My desire is to determine $u_3$ so that $\left \langle u_1,u_3 \right \rangle=\left \langle u_2,u_3 \right \rangle=0$

    $\left \langle u_1,u_3 \right \rangle=(1,2,1,3)*(v_1,v_2,v_3,v_4)=v_1+2v_2+v_3+3v_4=0$

    $\left \langle u_2,u_3 \right \rangle=(2,5,1,4)*(v_1,v_2,v_3,v_4)=2v_1+5v_2+v_3+4v_4=0$

    Thus I end up(after Gauss-elimination):

    $\begin{pmatrix}
    1 &0 &3 &7 \\
    0 &1 &-1 &-2
    \end{pmatrix}\begin{pmatrix}
    v_1\\
    v_2\\
    v_3\\
    v_4
    \end{pmatrix}=\begin{pmatrix}
    0\\
    0
    \end{pmatrix}$
    which has free variables $v_3,v_4$ but unable to solve.

    b) I know that the cross product of two vectors $a$ and $b$ results in a vector orthogonal to $a$ and $b$ that cannot be applied in $\mathbb R^4$. I was also recommended to use Gram-Schmidt but I don't know that yet. Is it more suitable for this problem?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 13, 2013 #2

    haruspex

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    You expect there to be two free variables at the end. Just leave those as parameters and write the general vector in terms of them.
     
  4. Apr 14, 2013 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    I wouldn't bother with matrices at all. Taking such a vector to be of the form (a, b, c, d) we have
    (a, b, c, d).(1, 2, 1, 3)= a+ 2b+ c+ 3d= 0 and
    (a, b, c, c).(2, 5, 1, 4)= 2a+ 5b+ c+ 4d= 0.

    Subtract the first equation from the second: a+ 3b+d= 0 so a= -3b- d.
    Replacing a by -3b+ d in the first equation, -3b+ d+ 2b+ c+ 3d= -b+ c+ 4d so c= b- 4d
     
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