find a vector u=(a,b,c) where a,b,c are not all 0 so that u is orthogonal to both x=(1,2,1) and y=(1,-1,1)
Can I use the cross product on x and y or would that be incorrect?
There is another method except cross product.
$\displaystyle
A=\begin{bmatrix} \vec{x} &\vec{y}
\end{bmatrix}~,
~\vec{u}\in im(A)^{\perp}=\ker(A^{T})
=\ker\begin{bmatrix}
1 &2 &1 \\
1 &-1 &1 \\
\end{bmatrix} \\
$
$\displaystyle
=\ker\begin{bmatrix}
1 &2 &1 \\
0 &-3 &0 \\
\end{bmatrix}
=\ker\begin{bmatrix}
1 &0 &1 \\
0 &1 &0 \\
\end{bmatrix}
=span( \begin{bmatrix}
-1 \\0 \\1
\end{bmatrix} ) \\
$
$\displaystyle
\rightarrow b=0,|a|=|c|\neq 0,a=-c
$
If dimension is high, cross product is complex