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Answered question

2021-11-22

Find two linearly independent vectors perpendicular to the vector

\vec{v} = [\begin{matrix} 1 \\ 3 \\ 9 \end{matrix}]

Answer & Explanation

Elizabeth Witte

Beginner2021-11-23Added 24 answers

To find linearly independent vector perpendicular to vector

\vec{v} = [\begin{matrix} 1 \\ 3 \\ 9 \end{matrix}]

solution
let

X = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}]

as we know, two vectors are perpendicular when

\vec{v} \cdot X = 0

substitute the values

[\begin{matrix} 1 & 3 & 9 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]

\Rightarrow x_{1} + 3 x_{2} + 9 x_{3} = 0

\Rightarrow x_{1} = - 3 x_{2} - 9 x_{3}

let

x_{2} = t, x_{3} = s

X = [\begin{matrix} - 3 t - 9 s \\ t \\ s \end{matrix}]

= [\begin{matrix} - 3 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} - 9 \\ 0 \\ 1 \end{matrix}]

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