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