The unit vector in the direction of $\vec{a}+\vec{b}$ if $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ & $\vec{b}=-\hat{i}+\hat{j}+-\hat{k}$ is :

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

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

$\vec{b}=-\hat{i}+\hat{j}-\bar{k}$

$\vec{a}+\vec{b}=\hat{i}+\hat{k}$

So unit vector in direction of $(\vec{a}+\vec{b})$

$=\frac{\vec{a}+\vec{b}}{|\vec{a}+\vec{b}|}=\frac{\hat{i}+\hat{k}}{\sqrt{1+1}}$

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