Question

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

• A. $\frac{\hat{i}+10\hat{j}-18\hat{k}}{5\sqrt{17}}$
• B. $\frac{\hat{i}-10\hat{j}+18\hat{k}}{5\sqrt{17}}$
• C. $\frac{\hat{i}-10\hat{j}-18\hat{k}}{5\sqrt{17}}$
• D. $\frac{\hat{i}+10\hat{j}+18\hat{k}}{5\sqrt{17}}$

Answer 1

The correct option is C $\frac{\hat{i}-10\hat{j}-18\hat{k}}{5\sqrt{17}}$

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