Explanation

(a)
Let $\overrightarrow{\mathbf{a}} = 2\widehat{\dot{\mathbf{i}}} + 4\widehat{\dot{\mathbf{j}}} - 5\widehat{\mathbf{k}}$, $\overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}}$
First diagonal, $\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} = 3\widehat{\dot{\mathbf{i}}} + 6\widehat{\dot{\mathbf{j}}} - 2\widehat{\mathbf{k}}$
$\Longrightarrow \left| \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} \right| = 7$
Second diagonal, $\overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} - 8\widehat{\mathbf{k}}$
$\Longrightarrow \left| \overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{b}} \right| = \sqrt{69}$