Explanation

(b)
Now, 2$\overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{c}} = 2\left( - \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}} \right) - \left( 2\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}} \right)$
$= \widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}}$
and $\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} = - \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}} + 2\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}}$
$= \widehat{\dot{\mathbf{i}}} + \widehat{\mathbf{k}}$let $\theta$ be the angle between $2\overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{c}}$ and $\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}}.$
$\therefore\cos{\theta = \frac{\left( \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right) \bullet \left( \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} \right)}{\sqrt{1^{2} + 1^{2}}\sqrt{1^{2} + 1^{2}}}}$
$\Longrightarrow \cos{\theta = \frac{1}{\sqrt{2}\sqrt{2}} = \frac{1}{2}}$
$\Longrightarrow \ \theta = \frac{\pi}{3}$