Question

Find a unit vector perpendicular to the vectors $\overrightarrow{A}=4\hat{i}-\hat{j}+3\hat{k}$ and $\overrightarrow{B}=-2\hat{i}+\hat{j}-2\hat{k}$.

Answer 1

Let $\overrightarrow{C}$=$\overrightarrow{A}\times \overrightarrow{B}$

Since we know cross product of two vectors produces another vector which is perpendicular to both the vectors.

$\overrightarrow{A}\times \overrightarrow{B}=\left[\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 4& -1& 3\\ -2& 1& -2\end{array}\right]\overrightarrow{A}\times \overrightarrow{B}=(2-3)\hat{i}-(-8+3)\hat{j}+(4-2)\hat{k}\overrightarrow{A}\times \overrightarrow{B}-\hat{i}+5\hat{j}+2\hat{k}$

Unit vector of $\overrightarrow{C}$=$\frac{\overrightarrow{C}}{\sqrt{\overrightarrow{C}}}$

$|\overrightarrow{C}|=\sqrt{1^2+{(-5)}^2+(2{)}^2}=\sqrt{28}=2\sqrt{7}$

$\overrightarrow{C}=\frac{\hat{i}-5\hat{j}+2\hat{k}}{2\sqrt{7}}$