Question

The acute angle between the line joining the points $ (2,1,-3),(-3,1,7) $ and a line parallel to $ \frac{x-1}{3}=\frac{y}{4}=\frac{z+3}{5} $ through die point $ (-1,0,4) $ is

• 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) $

Answer 1

Answer: (C)

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) $