Question

The acute angle between two lines such that the direction cosines l,m,n of each of them satisfy the equations $l+m+n=0$ and $l^2+m^2-n^2=0$ is :-

• A. $30$
• B. $45$
• C. $60$
• D. $15$

Answer 1

Answer: (C)

$60$

Lines are $l+m+n=0\Rightarrow -l=(m+n)$
and $l^2-m^2+n^2=0\Rightarrow l^2=m^2-n^2$
Solving them gives, ${(-\left(m+n\right))}^2=m^2+n^2+2mn=m^2-n^2\Rightarrow 2n(n+m)=0$
Now, for $n=0$
$m=-l$, so d.c's $(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0)$
And for $n=-m$
$l=0$ so d.c's $(0,\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$
Angle between the lines $\left|\cos \theta \right|=\left|\frac{1}{2}\right|\Rightarrow \theta =\frac{\pi }{3}=60$