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Q.
Direction ratios of two lines are $ 5,-12,13 $ and $ -3,4,5, $ then angle between the lines is
Rajasthan PETRajasthan PET 2001
Solution:
Direction ratios of both lines are
$ {{a}_{1}}=5,{{b}_{1}}=-12,{{c}_{1}}=13 $ and $ {{a}_{2}}=-3,{{b}_{2}}=4,{{c}_{2}}=5, $
then angle between them is given by
$ \cos \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}} $
$ =\frac{5\times -3+4\times -12+13\times 5}{\sqrt{{{(5)}^{2}}+{{(-12)}^{2}}+{{(13)}^{2}}}\sqrt{{{(-3)}^{2}}+{{(4)}^{2}}+{{(5)}^{2}}}} $
$ =\frac{-15-48+65}{\sqrt{25+144+169}\sqrt{9+16+25}} $
$ =\frac{2}{13\sqrt{2}.5\sqrt{2}} $
$ \Rightarrow $ $ \cos \theta =\frac{1}{65} $
$ \Rightarrow $ $ \theta ={{\cos }^{-1}}\left( \frac{1}{65} \right) $