Vectors in R^4 orthogonal to two vectors

1. Apr 13, 2013

zeralda21

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. Apr 13, 2013

haruspex

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.

3. Apr 14, 2013

HallsofIvy

Staff Emeritus
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