What is the unit vector perpendicular to the following vectors $2 \hat{i} + 2 \hat{j} - \hat{k}$ and $6 \hat{i} - 3 \hat{j} + 2 \hat{k}$

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$\frac{\hat{i} + 10 \hat{j} - 18 \hat{k}}{5 \sqrt{17}}$

$\frac{- \hat{i} - 10 \hat{j} + 18 \hat{k}}{5 \sqrt{17}}$

$\frac{\hat{i} - 10 \hat{j} - 18 \hat{k}}{5 \sqrt{17}}$

$\frac{\hat{i} + 10 \hat{j} + 18 \hat{k}}{5 \sqrt{17}}$

answer is C.

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Detailed Solution

$\begin{aligned} \vec{A} = 2 \hat{i} + 2 \hat{j} - \hat{k} and \vec{B} = 6 \hat{i} - 3 \hat{j} + 2 \hat{k} \\ \vec{C} = \vec{A} \times \vec{B} = (2 \hat{i} + 2 \hat{j} - \hat{k}) \times (6 \hat{i} - 3 \hat{j} + 2 \hat{k}) \end{aligned}$
$= | \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 2 & 2 & - 1 \\ 6 & - 3 & 2 \end{array} | = \hat{i} - 10 \hat{j} - 18 \hat{k}$
Unit vector perpendicular to both $\vec{A} and \vec{B}$
$= \frac{\hat{i} - 10 \hat{j} - 18 \hat{k}}{\sqrt{1^{2} + 10^{2} + 18^{2}}} = \frac{\hat{i} - 10 \hat{j} - 18 \hat{k}}{5 \sqrt{17}}$