Coplanar Lines
Category : JEE Main & Advanced
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 \[\left| \,\begin{matrix} x-{{x}_{2}} & y-{{y}_{2}} & z- {{z}_{2}} \\ {{l}_{1}} & {{m}_{1}} & {{n}_{1}} \\ {{l}_{2}} & {{m}_{2}} & {{n}_{2}} \\ \end{matrix}\, \right|=0\].
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