Explanation

(b)
[$\overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}}\ \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \ \overrightarrow{\mathbf{c}}\rbrack = \overrightarrow{\mathbf{a}} \bullet \{\left( \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} \right) \times \left( \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} \right)\}$
$= \overrightarrow{\mathbf{a}} \bullet \left\{ \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{b}} \right\}$
$= \overrightarrow{\mathbf{a}} \bullet \left\{ \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right\}$
$= \left\lbrack \ \overrightarrow{\mathbf{a}}\ \ \overrightarrow{\mathbf{b}}\ \ \overrightarrow{\mathbf{a}} \right\rbrack + \left\lbrack \ \overrightarrow{\mathbf{a}}\ \ \overrightarrow{\mathbf{c}}\ \ \overrightarrow{\mathbf{a}} \right\rbrack = 0$