Explanation

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