Explanation

(c)
Since, $\overrightarrow{\mathbf{a}} \bullet \left( \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} \right) = 0$
$\Longrightarrow \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} = 0\ \ \$….(i)
Similarly, $\overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} = 0\ \ \$…..(ii)
and $\overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} = 0$ ….(iii)
On adding Eqs. (i),(ii) and (iii), we get
$2\left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} \right) = 0$
Now, $|\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}}|^{2} = |\overrightarrow{\mathbf{a}}|^{2} + |\overrightarrow{\mathbf{b}}|^{2} + |\overrightarrow{\mathbf{c}}|^{2} + 2(\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}})$
$= |\overrightarrow{\mathbf{a}}|^{2} + |\overrightarrow{\mathbf{b}}|^{2} + |\overrightarrow{\mathbf{c}}|^{2}$
$= 9 + 16 + 25 = 50$
$\Longrightarrow \left| \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} \right| = 5\sqrt{2}$