A) \[{{\cos }^{-1}}\left( \frac{7}{5\sqrt{10}} \right)\]
B) \[{{\cos }^{-1}}\left( \frac{1}{\sqrt{10}} \right)\]
C) \[{{\cos }^{-1}}\left( \frac{3}{5\sqrt{10}} \right)\]
D) \[{{\cos }^{-1}}\left( \frac{1}{5\sqrt{10}} \right)\]
Correct Answer: A
Solution :
Direction ratio of the line joining the point \[(2,\,\,1,\,\,-3),\,\] \[\,(-\,3,\,\,1,\,\,7)\] are \[({{a}_{1}},\,\,{{b}_{1}},\,\,{{c}_{1}})\,\] \[\,\Rightarrow \,\,(-\,3-2,\,\,1-1,\,\,7-(-3))\,\,\Rightarrow \,\,(-\,5,\,\,0,\,\,10)\] Direction ratio of the line parallel to line \[\frac{x-1}{3}=\frac{y}{4}=\frac{z+3}{5}\] are \[({{a}_{2}},\,{{b}_{2}},\,\,{{c}_{2}})\,\,\Rightarrow \,\,(3,\,\,4,\,\,5)\] Angle between two lines, \[\cos \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\] \[\cos \theta =\frac{(-\,5\times 3)+(0\times 4)+(10\times 5)}{\sqrt{25+0+100}\sqrt{9+16+25}}\] \[\cos \theta =\frac{35}{25\sqrt{10}}\,\,\Rightarrow \,\,\theta ={{\cos }^{-1}}\left( \frac{7}{5\sqrt{10}} \right)\].
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