Question

The angle between the lines whose direction cosines satisfy the equations
$l+m+n=0$ and $l^2=m^2+n^2$, is

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

Answer 1

The correct option is A

$\frac{\pi }{3}$

We know that, angle between two lines is
$\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}l+m+n=0\Rightarrow l=-(m+n)\Rightarrow (m+n{)}^2=l^2\Rightarrow m^2+n^2+2mn=m^2+n^2[{\because }^2=m^2+n^2,given]\Rightarrow 2mn=0\text{When}m=0\Rightarrow l=-n\text{Hence},(l,m,n\left)is\right(1,0,-1).\text{When}n=0,\text{then}l=-m\text{Hence},(l,m,n\left)\text{is}\right(1,0,-1).\therefore \cos \theta =\frac{1+0+0}{\sqrt{2}\times \sqrt{2}}=\frac{1}{2}\Rightarrow \theta =\frac{\pi }{3}$