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Homework Help: 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):

    1 &0 &3 &7 \\
    0 &1 &-1 &-2
    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


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    Science Advisor
    Homework Helper
    Gold Member

    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


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    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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