The angle between two lines whose direction cosines satisfy $l+m+n=0$ and $l^2=m^2+n^2$ is
Step-by-Step Solution
Key Concept: $l+m+n=0$ and $l^2=m^2+n^2\Rightarrow(-m-n)^2=m^2+n^2\Rightarrow mn=0$
$(m+n)^2=m^2+n^2\Rightarrow 2mn=0\Rightarrow m=0$ or $n=0$. Case 1: $m=0$: $l=-n$, $l^2=n^2$ ✓. DC: $l=1/\sqrt{2},m=0,n=-1/\sqrt{2}$. Case 2: $n=0$: $l=-m$, $l^2=m^2$ ✓. DC: $l=1/\sqrt{2},m=-1/\sqrt{2},n=0$. $\cos\theta=(1/\sqrt{2})(1/\sqrt{2})+(0)(-1/\sqrt{2})+(-1/\sqrt{2})(0)=1/2\Rightarrow\theta=\pi/3$.
Correct Answer: 1