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?

Printable View

- Feb 8th 2010, 03:38 PMsuperdudefind 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? - Feb 8th 2010, 06:41 PMtonio
- Feb 9th 2010, 04:17 AMmath2009
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