Question

Write a unit vector in the direction of the sum of the vectors $\overrightarrow{a}=2\hat{i}+2\hat{j}-5\hat{k}$ and $\overrightarrow{b}=2\hat{i}+\hat{j}-7\hat{k}$. [CBSE 2014]

Answer 1

We have, $\overrightarrow{a}=2\hat{i}+2\hat{j}-5\hat{k}$ and $\overrightarrow{b}=2\hat{i}+\hat{j}-7\hat{k}$.

$\therefore \overrightarrow{a}+\overrightarrow{b}=\left(2\hat{i}+2\hat{j}-5\hat{k}\right)+\left(2\hat{i}+\hat{j}-7\hat{k}\right)=4\hat{i}+3\hat{j}-12\hat{k}$

$\Rightarrow \left|\overrightarrow{a}+\overrightarrow{b}\right|=\sqrt{4^2+3^2+{\left(-12\right)}^2}=\sqrt{16+9+144}=\sqrt{169}=13$

∴ Required unit vector = $\frac{\overrightarrow{a}+\overrightarrow{b}}{\left|\overrightarrow{a}+\overrightarrow{b}\right|}=\frac{4\hat{i}+3\hat{j}-12\hat{k}}{13}=\frac{4}{13}\hat{i}+\frac{3}{13}\hat{j}-\frac{12}{13}\hat{k}$