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Find a vector orthogonal to both of u=(-6,4,2) , v=(3,1,5)
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Solution:

Let $x=(x_1, x_2, x_3) $

be the vector orthgonal to both u and v .Then

$(x.u) = (x_1, x_2, x_3)(-6,4,2) =0 $

=$-6x_1+4x_2+2x_3=0$-----(1)

Similarly

$(x.v) = (x_1, x_2, x_3)(3,1,5)=0$

= $3x_1+x_2+5x_3=0$-----(2)

By using crammers rule

$\frac {x_1}{\begin{bmatrix} 4 & 2 \\ 1 & 5 \end{bmatrix}}$=$\frac {-x_2}{\begin{bmatrix} -6 & -2 \\ 3 & 5 \end{bmatrix}}$=$\frac {x_3}{\begin{bmatrix} -6 & 4 \\ 3 & 1 \end{bmatrix}}$

$\frac{x_1}{20-2}$=$\frac{-x}{-30-6}$=$\frac{x_3}{-6-12}$

$\frac{x_1}{18}$=$\frac{-x_2}{-36}$=$\frac{x_3}{-18}$

$\frac{x_1}{1}$=$\frac{x_2}{2}$=$\frac{x_3}{-1}$

hence (1,2,-1) is orthogonal to both vectors.

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