The direction cosines of three lines passing through the origin are \[{{l}_{1}},{{m}_{1}},{{n}_{1}};\,{{l}_{2}},{{m}_{2}},{{n}_{2}}\]and \[{{l}_{3}},{{m}_{3}},{{n}_{3}}\]. The lines will be coplanar, if

\[\left| \,\begin{matrix} {{l}_{1}} & {{m}_{2}} & {{n}_{3}} \\ {{l}_{2}} & {{m}_{3}} & {{n}_{1}} \\ {{l}_{3}} & {{m}_{1}} & {{n}_{2}} \\ \end{matrix}\, \right|=0\]

\[\left| \,\begin{matrix} {{l}_{1}} & {{n}_{1}} & {{m}_{1}} \\ {{l}_{2}} & {{n}_{2}} & {{m}_{2}} \\ {{l}_{3}} & {{n}_{3}} & {{m}_{3}} \\ \end{matrix}\, \right|=0\]

\[{{l}_{1}}{{l}_{2}}{{l}_{3}}+{{m}_{1}}{{m}_{2}}{{m}_{3}}+{{n}_{1}}{{n}_{2}}{{n}_{3}}=0\]

The direction cosines of three lines passing through the origin are \[

1.	The direction cosines of three lines passing through the origin are \[{{l}_{1}},{{m}_{1}},{{n}_{1}};\,{{l}_{2}},{{m}_{2}},{{n}_{2}}\]and \[{{l}_{3}},{{m}_{3}},{{n}_{3}}\]. The lines will be coplanar, if
A.	\[\left\| \,\begin{matrix} {{l}_{1}} & {{n}_{1}} & {{m}_{1}} \\ {{l}_{2}} & {{n}_{2}} & {{m}_{2}} \\ {{l}_{3}} & {{n}_{3}} & {{m}_{3}} \\ \end{matrix}\, \right\|=0\]
B.	\[\left\| \,\begin{matrix} {{l}_{1}} & {{m}_{2}} & {{n}_{3}} \\ {{l}_{2}} & {{m}_{3}} & {{n}_{1}} \\ {{l}_{3}} & {{m}_{1}} & {{n}_{2}} \\ \end{matrix}\, \right\|=0\]
C.	\[{{l}_{1}}{{l}_{2}}{{l}_{3}}+{{m}_{1}}{{m}_{2}}{{m}_{3}}+{{n}_{1}}{{n}_{2}}{{n}_{3}}=0\]
D.	None of these
Answer» B. \[\left\| \,\begin{matrix} {{l}_{1}} & {{m}_{2}} & {{n}_{3}} \\ {{l}_{2}} & {{m}_{3}} & {{n}_{1}} \\ {{l}_{3}} & {{m}_{1}} & {{n}_{2}} \\ \end{matrix}\, \right\|=0\]

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The direction cosines of three lines passing through the origin are \[{{l}_{1}},{{m}_{1}},{{n}_{1}};\,{{l}_{2}},{{m}_{2}},{{n}_{2}}\]and \[{{l}_{3}},{{m}_{3}},{{n}_{3}}\]. The lines will be coplanar, if

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