find a vector that has this characteristic

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February 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?
February 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
February 9th 2010, 04:17 AM
math2009

There is another method except cross product.

$<br /> A=\begin{bmatrix} \vec{x} &\vec{y}<br /> \end{bmatrix}~,<br /> ~\vec{u}\in im(A)^{\perp}=\ker(A^{T})<br /> =\ker\begin{bmatrix}<br /> 1 &2 &1 \\<br /> 1 &-1 &1 \\<br /> \end{bmatrix} \\<br />$

$<br /> =\ker\begin{bmatrix}<br /> 1 &2 &1 \\<br /> 0 &-3 &0 \\<br /> \end{bmatrix}<br /> =\ker\begin{bmatrix}<br /> 1 &0 &1 \\<br /> 0 &1 &0 \\<br /> \end{bmatrix}<br /> =span( \begin{bmatrix}<br /> -1 \\0 \\1<br /> \end{bmatrix} ) \\<br />$

$<br /> \rightarrow b=0,|a|=|c|\neq 0,a=-c<br /> <br />$

If dimension is high, cross product is complex

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