The acute angle between two lines whose direction s are given by $l+m+n=0$ and $l^2+m^2+n^2=0$ is

$\begin{array}{l}\mathrm{l}+\mathrm{m}+\mathrm{n}=0\dots \dots \dots \left(1\right)\\ \Rightarrow \mathrm{lm}=0\Rightarrow \mathrm{l}=0\Rightarrow \left(1\right)\mathrm{n}=-\mathrm{m}\\ \Rightarrow (\mathrm{l},\mathrm{m},\mathrm{n})=(0;\mathrm{m};-\mathrm{m})=(0,1,-1)\Rightarrow \mathrm{m}=0\\ \text{ 1) }\mathrm{n}=-\mathrm{l}\Rightarrow \mathrm{l}:\mathrm{m}:\mathrm{n}=\mathrm{l}:0:-\mathrm{l}=(1,0,-1)\\ \Rightarrow \cos \mathrm{\theta }=\frac{0+0+1}{\sqrt{1+1}\sqrt{1+1}}=\frac{1}{2}\Rightarrow \mathrm{\theta }=\frac{\mathrm{\pi }}{3}\end{array}$