Explanation

(b)
The lines $\overrightarrow{r} = \overrightarrow{a} + \lambda(\overrightarrow{b} \times \overrightarrow{c})$ and $\overrightarrow{r} = \overrightarrow{b} + \mu(\overrightarrow{c} \times \overrightarrow{a})$ pass through $\overrightarrow{a}$ and $\overrightarrow{b}$, respectively, and are parallel to the vectors $\overrightarrow{b} \times \overrightarrow{c}$ and $\overrightarrow{c} \times \overrightarrow{a}$, respectively. Therefore, they intersect if $\overrightarrow{a} - \overrightarrow{b},\ \overrightarrow{b} \times \overrightarrow{c}$ and $\overrightarrow{c} \times \overrightarrow{a}$ are coplanar and so
$(\overrightarrow{a} - \overrightarrow{b}).\left\{ \left( \overrightarrow{b} \times \overrightarrow{c} \right) \times (\overrightarrow{c} \times \overrightarrow{a}) \right\} = 0$
$\Rightarrow \left( \overrightarrow{a} - \overrightarrow{b} \right).\left( \left\lbrack \overrightarrow{b}\overrightarrow{c}\overrightarrow{a} \right\rbrack\overrightarrow{c} - \left\lbrack \overrightarrow{b}\overrightarrow{c}\overrightarrow{c} \right\rbrack\overrightarrow{a} \right) = 0$
$\Rightarrow \left( \left( \overrightarrow{a} - \overrightarrow{b} \right).\overrightarrow{c} \right)\left\lbrack \overrightarrow{b}\overrightarrow{c}\overrightarrow{a} \right\rbrack = 0$
$\Rightarrow \overrightarrow{a}.\overrightarrow{c} - \overrightarrow{b}.\overrightarrow{c} = 0\ \Rightarrow \ \overrightarrow{a}.\overrightarrow{c} = \overrightarrow{b}.\overrightarrow{c}$