Explanation

(b)
Given, $\left\lbrack \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}}\ \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{a}} \right\rbrack = \left\lbrack \overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}} \right\rbrack$
$\Longrightarrow 2\left\lbrack \overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}} \right\rbrack = \left\lbrack \overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}} \right\rbrack$
$= \left\lbrack \overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}} \right\rbrack = 0$
Hence,$\ \overrightarrow{\mathbf{a}},\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ are coplanar.