Question

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

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

Answer 1

Answer: (C)

$\frac{\pi }{3}$

$Wehavethatanglebetweentwolinesis$

$\begin{array}{c}\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{{a_1}^2+{b_1}^2+{c_1}^2}\sqrt{{a_2}^2+{b_2}^2+{c_2}^2}}\\ l+m+n=0\\ \Rightarrow l=-\left(m+n\right)\\ \Rightarrow \left(m^2+n^2\right)=l^2\\ \Rightarrow m^2+n^2+2mn=m^2+n^2\\ \left[l^2=m^2+n^{2}given\right]\\ \Rightarrow 2mn=0\end{array}$

$when$ $m=0\Rightarrow l=-n$

$Hence,$ $\left(l,m,n\right)is\left(1,0,-1\right)$

$When$ $n=0\Rightarrow l=-m$

$Hence$ $\left(l,m,n\right)is\left(1,0,-1\right)$

$\cos \theta =\frac{1+0+0}{\sqrt{2}\times \sqrt{2}}=\frac{1}{2}\Rightarrow \theta =\frac{\pi }{3}$