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