Explanation

(a)
Since, $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are mutually perpendicular unit vectors.
$\Rightarrow \left| \overrightarrow{\mathbf{a}} \right| = \left| \ \overrightarrow{\mathbf{b}} \right| = \left| \overrightarrow{\mathbf{c}} \right| = 1$
and $\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}}\mathbf{=}\overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{c}}\mathbf{=}\overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{a}} = 0$ …..(i)
Now, $\left| \overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}\mathbf{+}\overrightarrow{\mathbf{c}} \right|^{2} = \left( \mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}\mathbf{+}\overrightarrow{\mathbf{c}} \right) \bullet (\mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}\mathbf{+}\overrightarrow{\mathbf{c}})\$
$= \left| \overrightarrow{\mathbf{a}} \right|^{2} + \left| \overrightarrow{\mathbf{b}} \right|^{2} + \left| \overrightarrow{\mathbf{c}} \right|^{2} + 2\left( \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}}\mathbf{+ \ }\overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{c}}\mathbf{+}\overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{a}}\ \right)$
$= 1 + 1 + 1 + 0 = 3$ [from Eq. (i) ]
$\Rightarrow |\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}\mathbf{+}\overrightarrow{\mathbf{c}}| = \sqrt{3}$