Find a unit vector in the direction of the sum of the vectors  $\overline{a}=2\overline{i}+2\overline{j}-5\overline{k}$ and $\overline{b}=2\overline{i}+\overline{j}+3\overline{k}$

$\overline{a}=2\overline{i}+2\overline{j}-5\overline{k}$

$\overline{b}=2\overline{i}+\overline{j}+3\overline{k}$

$\overline{a}+\overline{b}=4\overline{i}+3\overline{j}-2\overline{k}$ ;

$|\overline{a}+\overline{b}|=\sqrt{16+9+4}=\sqrt{29}$

Required unit vector

$=\frac{\overline{a}+\overline{b}}{|\overline{a}+\overline{\overline{b}}|}\Rightarrow \frac{4\overline{i}+3\overline{j}-2\overline{k}}{\sqrt{29}}$