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 $\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$ $=\frac{(-5\times 3)+(0\times 4)+(10\times 5)}{\sqrt{25+0+100}\sqrt{9+16+25}}=\frac{35}{25\sqrt{10}}$ $\Rightarrow$ $\theta ={\cos }^{-1}\left(\frac{35}{25\sqrt{10}}\right)={\cos }^{-1}\left(\frac{7}{5\sqrt{10}}\right)$