# find a vector that has this characteristic

• Feb 8th 2010, 03:38 PM
superdude
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?
• Feb 8th 2010, 06:41 PM
tonio
Quote:

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
• Feb 9th 2010, 04:17 AM
math2009
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