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Question: If \(l _ { 1 } , m _ { 1 } , n _ { 1 }\) and \(l _ { 2 } , m _ { 2 } , n _ { 2 }\) are d.c.’s of t...

If l1,m1,n1l _ { 1 } , m _ { 1 } , n _ { 1 } and l2,m2,n2l _ { 2 } , m _ { 2 } , n _ { 2 } are d.c.’s of two lines inclined to each other at an angle θ, then the d.c.’s of the internal bisectors of angle between these lines are

A

l1+l22sin⁡θ/2,m1+m22sin⁡θ/2,n1+n22sin⁡θ/2\frac { l _ { 1 } + l _ { 2 } } { 2 \sin \theta / 2 } , \frac { m _ { 1 } + m _ { 2 } } { 2 \sin \theta / 2 } , \frac { n _ { 1 } + n _ { 2 } } { 2 \sin \theta / 2 }

B

l1+l22cos⁡θ/2,m1+m22cos⁡θ/2,n1+n22cos⁡θ/2\frac { l _ { 1 } + l _ { 2 } } { 2 \cos \theta / 2 } , \frac { m _ { 1 } + m _ { 2 } } { 2 \cos \theta / 2 } , \frac { n _ { 1 } + n _ { 2 } } { 2 \cos \theta / 2 }

C

l1−l22sin⁡θ/2,m1−m22sin⁡θ/2,n1−n22sin⁡θ/2\frac { l _ { 1 } - l _ { 2 } } { 2 \sin \theta / 2 } , \frac { m _ { 1 } - m _ { 2 } } { 2 \sin \theta / 2 } , \frac { n _ { 1 } - n _ { 2 } } { 2 \sin \theta / 2 }

D

l1−l22cos⁡θ/2,m1−m22cos⁡θ/2,n1−n22cos⁡θ/2\frac { l _ { 1 } - l _ { 2 } } { 2 \cos \theta / 2 } , \frac { m _ { 1 } - m _ { 2 } } { 2 \cos \theta / 2 } , \frac { n _ { 1 } - n _ { 2 } } { 2 \cos \theta / 2 }

Answer

l1+l22cos⁡θ/2,m1+m22cos⁡θ/2,n1+n22cos⁡θ/2\frac { l _ { 1 } + l _ { 2 } } { 2 \cos \theta / 2 } , \frac { m _ { 1 } + m _ { 2 } } { 2 \cos \theta / 2 } , \frac { n _ { 1 } + n _ { 2 } } { 2 \cos \theta / 2 }

Explanation

Solution

Let OA and OB be two lines.

D.c.’s of OA is (l1,m1,n1)\left( l _ { 1 } , m _ { 1 } , n _ { 1 } \right) and OB is (l2,m2,n2)\left( l _ { 2 } , m _ { 2 } , n _ { 2 } \right) .

Let OA = OB = 1.

Then the co-ordinates of A and B are (l1,m1,n1)\left( l _ { 1 } , m _ { 1 } , n _ { 1 } \right) and (l2,m2,n2)\left( l _ { 2 } , m _ { 2 } , n _ { 2 } \right)

Let OC be the bisector of ∠AOB.

Then C is the mid-point of AB and so its co-ordinates are (l1+l22,m1+m22,n1+n22)\left( \frac { l _ { 1 } + l _ { 2 } } { 2 } , \frac { m _ { 1 } + m _ { 2 } } { 2 } , \frac { n _ { 1 } + n _ { 2 } } { 2 } \right) .

∴ D.r.’s of line OC are (l1+l22,m1+m22,n1+n22)\left( \frac { l _ { 1 } + l _ { 2 } } { 2 } , \frac { m _ { 1 } + m _ { 2 } } { 2 } , \frac { n _ { 1 } + n _ { 2 } } { 2 } \right)

We have,

=12l12+m12+n12+l22+m22+n22+2(l1l2+m1m2+n1n2)= \frac { 1 } { 2 } \sqrt { l _ { 1 } ^ { 2 } + m _ { 1 } ^ { 2 } + n _ { 1 } ^ { 2 } + l _ { 2 } ^ { 2 } + m _ { 2 } ^ { 2 } + n _ { 2 } ^ { 2 } + 2 \left( l _ { 1 } l _ { 2 } + m _ { 1 } m _ { 2 } + n _ { 1 } n _ { 2 } \right) }

=121+1+2cos⁡θ=122(2⋅cos⁡2θ/2)=cos⁡θ/2= \frac { 1 } { 2 } \sqrt { 1 + 1 + 2 \cos \theta } = \frac { 1 } { 2 } \sqrt { 2 \left( 2 \cdot \cos ^ { 2 } \theta / 2 \right) } = \cos \theta / 2 .

D.r.’s of line OC are l1+l22cos⁡θ/2,m1+m22cos⁡θ/2,n1+n22cos⁡θ/2\frac { l _ { 1 } + l _ { 2 } } { 2 \cos \theta / 2 } , \frac { m _ { 1 } + m _ { 2 } } { 2 \cos \theta / 2 } , \frac { n _ { 1 } + n _ { 2 } } { 2 \cos \theta / 2 }