Explanation

(b)
$\therefore\ \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} - 3\widehat{\mathbf{k}} + 3\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}$
$= 4\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}}$
$and\ \overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{b}} = \left( \widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} - 3\widehat{\mathbf{k}} \right) - \left( 3\widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}} \right)$
$= - 2\widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} - 5\widehat{\mathbf{k}}$
$\therefore\ \cos{\theta = \frac{\left( \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} \right) \bullet (\overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{b}})}{|\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}}||\overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{b}}|}}$
$= \frac{\left( 4\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}} \right) \bullet ( - 2\widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} - 5\widehat{\mathbf{k}})}{|4\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}}|| - 2\widehat{\dot{\mathbf{i}}} + 3\widehat{\dot{\mathbf{j}}} - 5\widehat{\mathbf{k}}|}$
$= \frac{- 8 + 3 + 5}{\sqrt{16 + 1 + 1}\sqrt{4 + 9 + 25}} = 0$
$\Longrightarrow \ \theta = 90{^\circ}$