Question

Find perpendicular unit vector of vectors
$\hat{i}-2\hat{j}+\hat{k}$
and $2\hat{i}+\hat{j}-3\hat{k}$.

Answer 1

Let $\overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k}$
and $\overrightarrow{b}=2\hat{i}+\hat{j}-3\hat{k}$
Then perpendicular unit vector of $\overrightarrow{a}$ and $\overrightarrow{b}$,
$\hat{n}=\frac{\overrightarrow{a}\times \overrightarrow{b}}{|\overrightarrow{a}\times \overrightarrow{b}|}$
Now $\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -2& 1\\ 2& 1& -3\end{array}\right|$
$\Rightarrow \overrightarrow{a}\times \overrightarrow{b}=\hat{i}(6-1)-\hat{j}(-3-2)+\hat{k}(1+4)$
$\Rightarrow \overrightarrow{a}\times \overrightarrow{b}=5\hat{i}+5\hat{j}+5\hat{k}$
and $|\overrightarrow{a}\times \overrightarrow{b}|=\sqrt{(5{)}^2+(5{)}^2+(5{)}^2}$
$\Rightarrow |\overrightarrow{a}\times \overrightarrow{b}|=\sqrt{25+25+25}$
$\Rightarrow |\overrightarrow{a}\times \overrightarrow{b}|=\sqrt{75}$
$\Rightarrow |\overrightarrow{a}\times \overrightarrow{b}|=5\sqrt{3}$
$\therefore$ Perpendicular unit vector between $\overrightarrow{a}$ and $\overrightarrow{b}$,
$\hat{n}=\frac{5\hat{i}+5\hat{j}+5\hat{k}}{5\sqrt{3}}$
$=\frac{5(\hat{i}+\hat{j}+\hat{k})}{5\sqrt{3}}$
$=\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$
Hence the perpendicular unit vector between $\hat{i}-2\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}-3\hat{k}$ is $\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$