The equation of the plane containing the lines $\vec{r} = \vec{a} + λ\vec{b}$ and $\vec{r} = \vec{b} + \mu \vec{b}$ is

$[\vec{r} \vec{a} \vec{b}]= 0 $

The required plane passes through the points having position vectors $\vec{a}$ and $\vec{b}$. It is perpendicular to $\vec{a}×\vec{b}$.

So, the equation of the plane is

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