$n =2(\ell+ m )$
$\ell m + n (\ell+ m )=0$
$\ell m +2(\ell+ m )^{2}=0$
$2 \ell^{2}+2 m ^{2}+5 m \ell=0$
$2\left(\frac{\ell}{ m }\right)^{2}+2+5\left(\frac{\ell}{ m }\right)=0$
$2 t ^{2}+5 t +2=0$
$( t +2)(2 t +1)=0$
$\Rightarrow t =-2 ;-\frac{1}{2}$
(i) $\frac{\ell}{ m } =-2 $
$ \frac{ n }{ m } =-2 $
$(-2 m , m ,-2 m ) $
$(-2,1,-2) $
(ii) $\frac{\ell}{ m }=-\frac{1}{2}$
$n =-2 \ell$
$(\ell,-2 \ell,-2 \ell)$
$(1,-2,-2)$
$\cos \theta=\frac{-2-2+4}{\sqrt{9} \sqrt{9}}=0$
$ \Rightarrow 0=\frac{\pi}{2}$