Question

If dc's of two intersecting lines $(l,m,n)$ are connected by relations $l+m+n=0,lm=0,$ then angle between the lines is:
(where $n<0$)

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

Answer 1

The correct option is A $\frac{\pi }{3}$

Given: $l+m+n=0,lm=0,$
Since,$lm=0$
So either, $l=0orm=0$
When $l=0$
$0+m+n=0$
$n=-m$
$l^2+m^2+n^2=1$
$0+(m{)}^2+(-m{)}^2=1$
$2m^2=1$
$m=\frac{1}{\sqrt{2}}$
So, $n=-\frac{1}{\sqrt{2}}$
Thus $(l,m,n)=(0,\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$
Thus $(a_1,b_1,c_1)=(0,1,-1)=\overrightarrow{b_1}\cdots \left(i\right)$
When $m=0$
$l+0+n=0$
$n=-l$
$l^2+0^2+(-l{)}^2=1$
$l^2+l^2=1$
$2l^2=1$
$l=\frac{1}{\sqrt{2}}$
So, $n=-\frac{1}{\sqrt{2}}$
Thus $(l,m,n)=(\frac{1}{\sqrt{2}},0,-\frac{1}{\sqrt{2}})$
Thus, $(a_2,b_2,c_2)=(1,0,-1)=\overrightarrow{b_2}\cdots \left(ii\right)$
Angle between two lines: $\cos \theta =\frac{\overrightarrow{b_1}.\overrightarrow{b_2}}{|\overrightarrow{b_1}||\overrightarrow{b_2}|}$
$\cos \theta =\frac{0\times 1+1\times 0+(-1)\times (-1)}{\sqrt{0+1+1}\sqrt{1+0+1}}$
$\cos \theta =\frac{1}{2}$
$\theta =\frac{\pi }{3}$