Find the vectors whose magnitude is $5$ and which are perpendicular to the vectors $3 \hat{i} + \hat{j} - 4 \hat{k}$ and $6 \hat{i} + 5 \hat{j} - 2 \hat{k}$
Let $\;$ $\overrightarrow{a} = 3 \hat{i} + \hat{j} - 4 \hat{k}$; $\;$ $\overrightarrow{b} = 6 \hat{i} + 5 \hat{j} - 2 \hat{k}$
Let unit vector perpendicular to $\overrightarrow{a}$ and $\overrightarrow{b}$ be $\hat{n}$.
Then, $\hat{n} = \pm \left(\dfrac{\overrightarrow{a} \times \overrightarrow{b}}{\left|\overrightarrow{a} \times \overrightarrow{b}\right|}\right)$
$\begin{aligned}
\overrightarrow{a} \times \overrightarrow{b} & = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
3 & 1 & -4 \\
6 & 5 & -2
\end{vmatrix} \\\\
& = \hat{i} \left(-2 + 20\right) - \hat{j} \left(-6 + 24\right) + \hat{k} \left(15 - 6\right) \\\\
& = 18 \hat{i} - 18 \hat{j} + 9 \hat{k}
\end{aligned}$
$\left|\overrightarrow{a} \times \overrightarrow{b}\right| = \sqrt{\left(18\right)^2 + \left(-18\right)^2 + \left(9\right)^2} = \sqrt{729} = 27$
$\therefore$ $\;$ $\hat{n} = \pm \left(\dfrac{18 \hat{i} - 18 \hat{j} + 9 \hat{k}}{27}\right) = \pm \left(\dfrac{2 \hat{i} - 2 \hat{j} + \hat{k}}{3}\right)$
$\therefore$ $\;$ Vectors of magnitude $5$ perpendicular to $\overrightarrow{a}$ and $\overrightarrow{b}$
$= \pm 5 \times \left(\dfrac{2 \hat{i} - 2 \hat{j} + \hat{k}}{3}\right) = \pm \left(\dfrac{10 \hat{i} - 10 \hat{j} + 5 \hat{k}}{3}\right)$