Question

For given vectors $\overrightarrow{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $b=-\hat{i}+\hat{j}-\hat{k}$, then a unit vector in the direction of the vector $\overrightarrow{a}+\overrightarrow{b}$ is

• A. $\hat{i}+\hat{j}-2\hat{k}$
• B. $\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{k}$
• C. $\sqrt{2\hat{i}}+\sqrt{2}\hat{k}$
• D. None

Answer 1

Answer: (B)

$\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{k}$

Now $\overrightarrow{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{b}=-\hat{i}+\hat{j}-\hat{k}$
Thus $\overrightarrow{a}+\overrightarrow{b}=(2\hat{i}-\hat{j}+2\hat{k})+(\hat{i}+\hat{j}-\hat{k})=\hat{i}+\hat{k}$
Now $|\overrightarrow{a}+\overrightarrow{b}|=\sqrt{1^2+1^2}=\sqrt{2}$
Hence, unit vector along $\overrightarrow{a}=\frac{\overrightarrow{a}+\overrightarrow{b}}{|\overrightarrow{a}+\overrightarrow{b}|}$
$=\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{k}$