Vectors in R^4 orthogonal to two vectors

[zeralda21]

Homework Statement

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

Homework Equations

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

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?

 

[haruspex]

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.

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.

 

[HallsofIvy]

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