A) \[\frac{\pi }{3}\]
B) \[{{\tan }^{-1}}\left( \frac{1}{5} \right)\]
C) \[{{\cos }^{-1}}\left( \frac{1}{3} \right)\]
D) \[\frac{\pi }{2}\]
Correct Answer: C
Solution :
Given lines are \[\frac{x-5}{-3}=\frac{y+3}{-4}=\frac{z-7}{0}\] here, \[<{{l}_{1}},\,{{m}_{1}},\,{{n}_{1}}>\,\,=\,<\,-3,\,-4,0>\] and \[\frac{x}{1}=\frac{y-1}{-2}=\frac{z-6}{2}\] here, \[<{{l}_{2}},{{m}_{2}},{{n}_{2}}>\,=\,<1,\,-2,\,2>\] let \[\theta \] be the angle between the lines. Then, \[\cos \,\theta =\left| \frac{{{l}_{1}}{{l}_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}}}{\sqrt{l_{1}^{2}+m_{1}^{2}+n_{1}^{2}}+\sqrt{l_{2}^{2}+m_{2}^{2}+n_{2}^{2}}} \right|\] \[=\left| \frac{(-3)(1)+(-4)(-2)+(0)(2)}{\sqrt{9+16+0}\,\sqrt{1+4+4}} \right|\] \[=\left| \frac{-3+8+0}{5\times 3} \right|=\left| \frac{5}{5\times 3} \right|=\frac{1}{3}\] \[\therefore \] \[\theta ={{\cos }^{-1}}\,\left( \frac{1}{3} \right)\]
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