Question

A vector of magnitude 5 and perpendicular to $\hat{i}-2\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}-3\hat{k}$ is

• A. $\frac{5\sqrt{3}}{3}\left(\hat{i}+\hat{j}-\hat{k}\right)$
• B. $\frac{5\sqrt{3}}{3}\left(\hat{i}+\hat{j}+\hat{k}\right)$
• C. $\frac{5\sqrt{3}}{3}\left(\hat{i}-\hat{j}+\hat{k}\right)$
• D. $\frac{5\sqrt{3}}{3}\left(-\hat{i}+\hat{j}+\hat{k}\right)$

Answer 1

The correct option is B $\frac{5\sqrt{3}}{3}\left(\hat{i}+\hat{j}+\hat{k}\right)$

Let the vectors

$\begin{array}{c}\overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k}\\ \overrightarrow{b}=2\hat{i}+\hat{j}-3\hat{k}\end{array}$

and the vector perpendicular to $a$ and $b$ let it is $c$.

Now, for perpendicular

$\overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}$

$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -2& 1\\ 2& 1& -3\end{array}\right|$

$=5\hat{i}+5\hat{j}+5\hat{k}$

$\left|\overrightarrow{a}\times \overrightarrow{b}\right|=\sqrt{25+25+25}=5\sqrt{3}$

Unit vector along $c$ is $\frac{1}{5\sqrt{3}}(5\hat{i}+5\hat{j}+5\hat{k})=\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$

and and vector of magnitude $5$

$=\frac{5}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})=\frac{5\sqrt{3}}{3}\left(\hat{i}+\hat{j}+\hat{k}\right)$