Question

Find a unit vector perpendicular to both the vectors
$(2\hat{i}+3\hat{j}+\hat{k})$ and $(\hat{i}-\hat{j}+2\hat{k})$

• A. $\frac{1}{\sqrt{8}}(4\hat{i}+3\hat{j}-5\hat{k})$
• B. $\frac{1}{\sqrt{83}}(7\hat{i}-3\hat{j}-5\hat{k})$
• C. $\frac{1}{\sqrt{8}}(4\hat{i}+3\hat{j}+1\hat{k})$
• D. None of these

Answer 1

Answer: (B)

Let $\overrightarrow{A}=2\hat{i}+3\hat{j}+\hat{k}$ and
$\overrightarrow{B}=\hat{i}-\hat{j}+2\hat{k}$ unit vector
perpendicular to both $\overrightarrow{A}$ and
$\overrightarrow{B}$ is $n=\frac{\overrightarrow{A}\times \overrightarrow{B}}{|\overrightarrow{A}\times \overrightarrow{B}|}$
$\overrightarrow{A}\times \overrightarrow{B}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& 1\\ 1& -1& 2\end{array}\right|$
$=\hat{i}(6+1)-\hat{j}(4-1)+\hat{k}(-2-3)$
$=7\hat{i}-3\hat{j}-5\hat{k}$
$\therefore |\overrightarrow{A}\times \overrightarrow{B}|=\sqrt{7^2+(-3{)}^2+(-5{)}^2}$
$=\sqrt{83}unit$
$\therefore \hat{n}=\frac{1}{\sqrt{83}}$
$(7\hat{i}-3\hat{j}-5\hat{k})$