Explanation

(b)
$\left| \overrightarrow{\mathbf{OA}} \right| = \sqrt{4 + 4 + 1} = 3$
and| $\overrightarrow{\mathbf{OB}}| = \sqrt{4 + 16 + 16} = 6$
$\therefore\$ Required vector $= \lambda\left( \overrightarrow{\mathbf{OA}} + \overrightarrow{\mathbf{OB}} \right)$
$= \lambda\left\lbrack \frac{1}{3}\left( 2\widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right) + \frac{1}{6}(2\widehat{\dot{\mathbf{i}}} + 4\widehat{\dot{\mathbf{j}}} + 4\widehat{\mathbf{k}}) \right\rbrack$
$= \frac{\lambda}{3}\left( 3\widehat{\dot{\mathbf{i}}} + 4\widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}} \right)$
$\therefore Length\ of\ vector = \frac{\lambda}{3}\sqrt{9 + 16 + 9} = \frac{\lambda}{3}\sqrt{34}$
$Take\ \lambda = 2$
$\therefore Required\ length\ of\ a\ vector\ is\ \frac{\sqrt{136}}{3}$