# Find two linearly independent vectors perpendicular to the vector $\vec{v}=\begin{bmatrix}1\\3\\9\end{bmatrix}$

Find two linearly independent vectors perpendicular to the vector
$\vec{v}=\begin{bmatrix}1\\3\\9\end{bmatrix}$

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Elizabeth Witte
To find linearly independent vector perpendicular to vector
$\vec{v}=\begin{bmatrix}1\\3\\9\end{bmatrix}$
solution
let $X=\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}$
as we know, two vectors are perpendicular when $$\displaystyle\vec{{{v}}}\cdot{X}={0}$$
substitute the values
$\begin{bmatrix}1&3&9\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$
$$\displaystyle\Rightarrow{x}_{{1}}+{3}{x}_{{2}}+{9}{x}_{{3}}={0}$$
$$\displaystyle\Rightarrow{x}_{{1}}=-{3}{x}_{{2}}-{9}{x}_{{3}}$$
let $$\displaystyle{x}_{{2}}={t},\ {x}_{{3}}={s}$$
$X=\begin{bmatrix}-3t-9s\\t\\s\end{bmatrix}$
$=\begin{bmatrix}-3\\1\\0\end{bmatrix},\begin{bmatrix}-9\\0\\1\end{bmatrix}$