The length of the perpendicular from the origin to the plane passing through the point $\vec{a}$ and containing the line $\vec{r}= \vec{b} + λ \vec{c}$, is

$\frac{[\vec{a}\, \vec{b}\, \vec{c}]}{|\vec{b}×\vec{c}+\vec{c}×\vec{a}|}$

The plane passing through $\vec{a}$ and containing the line $\vec{r} = \vec{b} + λ \vec{c}$ also passes through the point $\vec{b}$ and is parallel to the vector $\vec{c}$. So, it is normal to the  vector $(\vec{a} - \vec{b})× \vec{c}$.

Thus, the equation of the plane is

$(\vec{r}-\vec{a}) .\begin{Bmatrix} (\vec{a} - \vec{b})× \vec{c}\end{Bmatrix}= 0 $

$⇒ (\vec{r}-\vec{a}) . ( \vec{a} × \vec{c}- \vec{b} × \vec{c}) = 0 $

$⇒\vec{r} . ( \vec{a} × \vec{c}- \vec{b} × \vec{c})= \vec{a}. ( \vec{a} × \vec{c}- \vec{b} × \vec{c})$

$⇒\vec{r} . ( \vec{a} × \vec{c}- \vec{b} × \vec{c})= -\vec{a}.(\vec{b} × \vec{c})$

$⇒\vec{r} . (\vec{b} × \vec{c}+ \vec{c} × \vec{a})- [\vec{a} \, \, \vec{b}\,\, \vec{c}]= 0 $

∴ Length of the perpendicular from the origin to this plane

$=\begin{vmatrix} \frac{\vec{0}.(\vec{b} × \vec{c}+ \vec{c} × \vec{a})- [\vec{a} \, \, \vec{b}\,\, \vec{c}]}{|\vec{b} × \vec{c}+ \vec{c} × \vec{a}|}\end{vmatrix}= \frac{[\vec{a}\, \vec{b}\, \vec{c}]}{|\vec{b}×\vec{c}+\vec{c}×\vec{a}|}$