Explanation

(a)
$\left( \overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{d}} \right) \times \left( \overrightarrow{\mathbf{b}} - \overrightarrow{\mathbf{c}} \right)$
$= \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} - \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}} - \overrightarrow{\mathbf{d}} \times \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{d}} \times \overrightarrow{\mathbf{c}}$
$= \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{d}} - \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{d}} - \overrightarrow{\mathbf{d}} \times \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{d}} \times \overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{0}}$
$\lbrack\because\ \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} = \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{d}},\ \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{d}},$ given]
$\Longrightarrow \left( \overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{d}} \right)||(\overrightarrow{\mathbf{b}} - \overrightarrow{\mathbf{c}})$
$\Longrightarrow \ \overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{d}} = \lambda(\overrightarrow{\mathbf{b}} - \overrightarrow{\mathbf{c}})$