Question

Find the unit vectors perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$, when $\overrightarrow{a}=3\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{b}=2\hat{i}+3\hat{j}-\hat{k}$

Answer 1

Consider the problem.

The vector which is perpendicular to both is given by cross product of two vector.

And

$\overrightarrow{a}=3\hat{i}+\hat{j}-2\hat{k}$

$\overrightarrow{b}=2\hat{i}+3\hat{j}-\hat{k}$

Then,

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

$\overrightarrow{c}=5\hat{i}-\hat{j}+7\hat{k}$

And, the unit in the direction of $\overrightarrow{c}$

$\hat{c}=\frac{\overrightarrow{c}}{|\overrightarrow{c}|}$ ---- (i)

And

$|\overrightarrow{c}|=\sqrt{(5{)}^2+(1{)}^2+(7{)}^2}=\sqrt{25+1+49}$

$=\sqrt{75}$

Therefore from (i)

$=\frac{5\hat{i}-\hat{j}+7\hat{k}}{\sqrt{75}}$

$=\left(\frac{5}{\sqrt{75}}\right)\hat{i}-\left(\frac{1}{\sqrt{75}}\right)\hat{j}+\left(\frac{7}{\sqrt{75}}\right)\hat{k}$

So unit vector which perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$