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

balff1t 2021-11-22 Answered
Find two linearly independent vectors perpendicular to the vector
\[\vec{v}=\begin{bmatrix}1\\3\\9\end{bmatrix}\]

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Expert Answer

Elizabeth Witte
Answered 2021-11-23 Author has 642 answers
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 \(\displaystyle{x}_{{2}}={t},\ {x}_{{3}}={s}\)
\[X=\begin{bmatrix}-3t-9s\\t\\s\end{bmatrix}\]
\[=\begin{bmatrix}-3\\1\\0\end{bmatrix},\begin{bmatrix}-9\\0\\1\end{bmatrix}\]
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