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Question: The direction cosines of three lines passing through the origin are \(l _ { 1 } , m _ { 1 } , n _ { ...

The direction cosines of three lines passing through the origin are l1,m1,n1;l2,m2,n2l _ { 1 } , m _ { 1 } , n _ { 1 } ; l _ { 2 } , m _ { 2 } , n _ { 2 }and l3,m3,n3l _ { 3 } , m _ { 3 } , n _ { 3 }. The lines will be coplanar, if

A

∣l1n1m1l2n2m2l3n3m3∣=0\left| \begin{array} { l l l } l _ { 1 } & n _ { 1 } & m _ { 1 } \\ l _ { 2 } & n _ { 2 } & m _ { 2 } \\ l _ { 3 } & n _ { 3 } & m _ { 3 } \end{array} \right| = 0

B

∣l1m2n3l2m3n1l3m1n2∣=0\left| \begin{array} { l l l } l _ { 1 } & m _ { 2 } & n _ { 3 } \\ l _ { 2 } & m _ { 3 } & n _ { 1 } \\ l _ { 3 } & m _ { 1 } & n _ { 2 } \end{array} \right| = 0

C

l1l2l3+m1m2m3+n1n2n3=0l _ { 1 } l _ { 2 } l _ { 3 } + m _ { 1 } m _ { 2 } m _ { 3 } + n _ { 1 } n _ { 2 } n _ { 3 } = 0

D

None of these

Answer

∣l1n1m1l2n2m2l3n3m3∣=0\left| \begin{array} { l l l } l _ { 1 } & n _ { 1 } & m _ { 1 } \\ l _ { 2 } & n _ { 2 } & m _ { 2 } \\ l _ { 3 } & n _ { 3 } & m _ { 3 } \end{array} \right| = 0

Explanation

Solution

Here, three given lines are coplanar if they have common perpendicular

Let d.c.'s of common perpendicular be l,m,nl , m , n

⇒ …..(i)

ll2+mm2+nn2=0l l _ { 2 } + m m _ { 2 } + n n _ { 2 } = 0 …..(ii)

and …..(iii)

Solving (ii) and (iii), we get

lm2n3−n2m3=mn2l3−n3l2=nl2m3−l3m2=k\frac { l } { m _ { 2 } n _ { 3 } - n _ { 2 } m _ { 3 } } = \frac { m } { n _ { 2 } l _ { 3 } - n _ { 3 } l _ { 2 } } = \frac { n } { l _ { 2 } m _ { 3 } - l _ { 3 } m _ { 2 } } = k

⇒ l=k(m2n3−n2m3),m=k(n2l3−n3l2),n=k(l2m3−l3m2)l = k \left( m _ { 2 } n _ { 3 } - n _ { 2 } m _ { 3 } \right) , m = k \left( n _ { 2 } l _ { 3 } - n _ { 3 } l _ { 2 } \right) , n = k \left( l _ { 2 } m _ { 3 } - l _ { 3 } m _ { 2 } \right)

Substituting in (i), we get

l1(m2n3−n2m3)+m1(n2l3−n3l2)+n1(l2m3−l3m2)=0l _ { 1 } \left( m _ { 2 } n _ { 3 } - n _ { 2 } m _ { 3 } \right) + m _ { 1 } \left( n _ { 2 } l _ { 3 } - n _ { 3 } l _ { 2 } \right) + n _ { 1 } \left( l _ { 2 } m _ { 3 } - l _ { 3 } m _ { 2 } \right) = 0

⇒\Rightarrow ∣l1m1n1l2m2n2l3m3n3∣=0\left| \begin{array} { l l l } l _ { 1 } & m _ { 1 } & n _ { 1 } \\ l _ { 2 } & m _ { 2 } & n _ { 2 } \\ l _ { 3 } & m _ { 3 } & n _ { 3 } \end{array} \right| = 0 ⇒\Rightarrow – ∣l1n1m1l2n2m2l3n3m3∣=0\left| \begin{array} { l l l } l _ { 1 } & n _ { 1 } & m _ { 1 } \\ l _ { 2 } & n _ { 2 } & m _ { 2 } \\ l _ { 3 } & n _ { 3 } & m _ { 3 } \end{array} \right| = 0 .