JEE Main & Advanced Mathematics Three Dimensional Geometry Coplanar Lines

Lines are said to be coplanar if they lie in the same plane or a plane can be made to pass through them.

Condition for the lines to be coplanar:

If the lines \[\frac{x-{{x}_{1}}}{{{l}_{1}}}=\frac{y-{{y}_{1}}}{{{m}_{1}}}=\frac{z-{{z}_{1}}}{{{n}_{1}}}\] and \[\frac{x-{{x}_{2}}}{{{l}_{2}}}=\] \[\frac{y-{{y}_{2}}}{{{m}_{2}}}=\] \[\frac{z-{{z}_{2}}}{{{n}_{2}}}\] are coplanar, then \[\left| \,\begin{matrix} {{x}_{2}}-{{x}_{1}} & {{y}_{2}}-{{y}_{1}} & {{z}_{2}}-{{z}_{1}}  \\ {{l}_{1}} & {{m}_{1}} & {{n}_{1}}  \\ {{l}_{2}} & {{m}_{2}} & {{n}_{2}}  \\ \end{matrix}\, \right|=0\].

The equation of the plane containing them is  \[\left| \,\begin{matrix} x-{{x}_{1}} & y-{{y}_{1}} & z-{{z}_{1}}  \\ {{l}_{1}} & {{m}_{1}} & {{n}_{1}}  \\ {{l}_{2}} & {{m}_{2}} & {{n}_{2}}  \\ \end{matrix}\, \right|=0\] or \[\left| \,\begin{matrix} x-{{x}_{2}} & y-{{y}_{2}} & z- {{z}_{2}}  \\ {{l}_{1}} & {{m}_{1}} & {{n}_{1}}  \\ {{l}_{2}} & {{m}_{2}} & {{n}_{2}}  \\ \end{matrix}\, \right|=0\].