Question

The angle between the lines $\frac{x-7}{1}=\frac{y+3}{-5}=\frac{z}{3}$ and $\frac{2-x}{-7}=\frac{y}{2}=\frac{z+1}{5}$ is equal to :

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{6}$
• E. $0$

Answer 1

The correct option is C $\frac{\pi }{2}$

Given lines are $\frac{x-7}{1}=\frac{y+3}{-5}=\frac{z}{3}$ ...(i)
and $\frac{2-x}{-7}=\frac{y}{2}=\frac{z+5}{1}$
$\Rightarrow \frac{x-2}{7}=\frac{y}{2}=\frac{z+5}{1}$ ...(ii)
DR's of line equations (i) and (ii) are
$(a_1,b_1,c_1)=(1,-5,3)$
and $(a_2,b_2,c_2)=(7,2,1)$
Therefore, $\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$
$=\frac{1\times 7+(-5)\times 2+3\times 1}{\sqrt{1^2+(-5{)}^2+(3{)}^2}\sqrt{(7{)}^2+(2{)}^2+(1{)}^2}}$
$=\frac{7-10+3}{\sqrt{1+25+9}\sqrt{49+4+1}}=0$
$\Rightarrow \theta =\frac{\pi }{2}$