If  l1,m1,n1and l2,m2,n2 are the direction cosines of the two lines inclined to each other at an angle θ , then the direction cosines of the external angular bisector of the angle between these lines are l1−l22 sinα,m1−m22 sinα,n1−n22 sinα where2α=

# If  ${l}_{1},{m}_{1},{n}_{1}$and ${l}_{2},{m}_{2},{n}_{2}$ are the direction cosines of the two lines inclined to each other at an angle $\theta$ , then the direction cosines of the external angular bisector of the angle between these lines are $\frac{{l}_{1}-{l}_{2}}{2\text{\hspace{0.17em}}\mathrm{sin}\alpha },\frac{{m}_{1}-{m}_{2}}{2\text{\hspace{0.17em}}\mathrm{sin}\alpha },\frac{{n}_{1}-{n}_{2}}{2\text{\hspace{0.17em}}\mathrm{sin}\alpha }$ where$2\alpha =$

1. A

$\theta$

2. B

$\frac{\theta }{2}$

3. C

$\frac{\theta }{4}$

4. D

None

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### Solution:

Given that the direction cosines of two lines are ${l}_{1},{m}_{1},{n}_{1}$ and${l}_{2},{m}_{2},{n}_{2}$

Given  $\theta$ is the acute angle between the lines then $\mathrm{cos}\theta ={l}_{1}{l}_{2}+{m}_{1}{m}_{2}+{n}_{1}{n}_{2}$

The direction ratios of angular bisector of the two lines whose direction cosines are ${l}_{1},{m}_{1},{n}_{1}$ and ${l}_{2},{m}_{2},{n}_{2}$ are $〈{l}_{1}-{l}_{2},{m}_{1}-{m}_{2},{n}_{1}-{n}_{2}〉$

Consider the value of

$\begin{array}{c}{\left({l}_{1}-{l}_{2}\right)}^{2}+{\left({m}_{1}-{m}_{2}\right)}^{2}+{\left({n}_{1}-{n}_{2}\right)}^{2}={{l}_{1}}^{2}+{{m}_{1}}^{2}+{{n}_{1}}^{2}+{{l}_{2}}^{2}+{{m}_{2}}^{2}+{{n}_{2}}^{2}+2{l}_{1}{l}_{2}+2{m}_{1}{m}_{2}+2{n}_{1}{n}_{2}\\ =2-2\left(\mathrm{cos}\theta \right)\\ =2\left(2{\mathrm{sin}}^{2}\frac{\theta }{2}\right)\\ =4{\mathrm{sin}}^{2}\frac{\theta }{2}\end{array}$

It implies that$\sqrt{{\left({l}_{1}-{l}_{2}\right)}^{2}+{\left({m}_{1}-{m}_{2}\right)}^{2}+{\left({n}_{1}-{n}_{2}\right)}^{2}}=2\mathrm{sin}\frac{\theta }{2}$

The direction cosines of angular bisector of given two lines are $\frac{{l}_{1}-{l}_{2}}{2sin\frac{\theta }{2}}=\frac{{m}_{1}-{m}_{2}}{2sin\frac{\theta }{2}}=\frac{{n}_{1}-{n}_{2}}{2sin\frac{\theta }{2}}$

Comparing the above direction cosines with $\frac{{l}_{1}-{l}_{2}}{2\text{\hspace{0.17em}}\mathrm{sin}\alpha },\frac{{m}_{1}-{m}_{2}}{2\text{\hspace{0.17em}}\mathrm{sin}\alpha },\frac{{n}_{1}-{n}_{2}}{2\text{\hspace{0.17em}}\mathrm{sin}\alpha }$, we have $\alpha =\frac{\theta }{2}$.

It implies that $2\alpha =\overline{)\theta }$

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