Explanation

(c)
The given plane passes through $\overrightarrow{a}$ and is parallel to the vectors $\overrightarrow{b} - \overrightarrow{a}$ and $\overrightarrow{c}$. So it is normal to $(\overrightarrow{b}–\overrightarrow{a}) \times \overrightarrow{c}$. Hence, its equation is
$\left( \overrightarrow{r} - \overrightarrow{a} \right) \bullet \left( \left( \overrightarrow{b} - \overrightarrow{a} \right) \times \overrightarrow{c} \right) = 0$
Or $\overrightarrow{r} \bullet \left( \overrightarrow{b} \times \overrightarrow{c} + \overrightarrow{c} \times \overrightarrow{a} \right) = \lbrack\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\rbrack$
The length of the perpendicular from the origin to this plane is
$\frac{\left\lbrack \overrightarrow{a}\overrightarrow{b}\overrightarrow{c} \right\rbrack}{\left| \overrightarrow{b} \times \overrightarrow{c} + \overrightarrow{c} \times \overrightarrow{a} \right|}$