Explanation

(c)
Let $Q(\overrightarrow{q})$ be the foot of altitude drawn from
$P\left( \overrightarrow{p} \right)$ to the line $\overrightarrow{r} = \overrightarrow{a} + \lambda\overrightarrow{b}$,
$\Rightarrow \ \left( \overrightarrow{q} - \overrightarrow{p} \right) \bullet \overrightarrow{b} = 0$ and $\overrightarrow{q} = \overrightarrow{a} + \lambda\overrightarrow{b}$
$\Rightarrow \left( \overrightarrow{a} + \lambda\overrightarrow{b} - \overrightarrow{p} \right) \bullet \overrightarrow{b} = 0$
$\Rightarrow \left( \overrightarrow{a} - \overrightarrow{p} \right) \bullet \overrightarrow{b} + \lambda\left| \overrightarrow{b} \right|^{2} = 0$
$\Rightarrow \ \lambda = \ \frac{\left( \overrightarrow{p} - \overrightarrow{a} \right) \bullet \overrightarrow{b}}{\left| \overrightarrow{b} \right|^{2}}$
$\Rightarrow \ \overrightarrow{q} - \overrightarrow{p} = \overrightarrow{a} + \frac{(\left( \overrightarrow{p} - \overrightarrow{a} \right) \bullet \overrightarrow{b})\overrightarrow{b}}{\left| \overrightarrow{b} \right|^{2}} - \overrightarrow{p}$
$\Rightarrow \left| \overrightarrow{q} - \overrightarrow{p} \right| = \left| \left( \overrightarrow{a} - \overrightarrow{p} \right) + \frac{(\left( \overrightarrow{p} - \overrightarrow{a} \right) \bullet \overrightarrow{b})\overrightarrow{b}}{\left| \overrightarrow{b} \right|^{2}} \right|$