# Thread: find a vector that has this characteristic

1. ## find a vector that has this characteristic

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?

2. Originally Posted by superdude
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?

Of course you can cross multiply the vectors and that's correct: the cross product is a vector orthogonal to both original ones.

Tonio

3. There is another method except cross product.

$
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} \\
$

$
=\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} ) \\
$

$
\rightarrow b=0,|a|=|c|\neq 0,a=-c

$

If dimension is high, cross product is complex