$\text{ Three distinct points }\mathrm{P}\left(3{\mathrm{u}}^2,2{\mathrm{u}}^3\right),\mathrm{Q}\left(3{\mathrm{v}}^2,2{\mathrm{v}}^3\right)\text{ and }$$\mathrm{R}\left(3{\mathrm{w}}^2,2{\mathrm{w}}^3\right)\text{ are collinear then - }$

$\left|\begin{array}{ccc}3{\mathrm{u}}^2& 2{\mathrm{u}}^3& 1\\ 3{\mathrm{v}}^2& 2{\mathrm{v}}^3& 1\\ 3{\mathrm{w}}^2& 2{\mathrm{w}}^3& 1\end{array}\right|=0$

${\mathrm{R}}_1\to {\mathrm{R}}_1-{\mathrm{R}}_2\text{ and }{\mathrm{R}}_2\to {\mathrm{R}}_2-{\mathrm{R}}_3$

$\Rightarrow \left|\begin{array}{ccc}\mathrm{u}+\mathrm{v}& {\mathrm{u}}^2+{\mathrm{v}}^2+\mathrm{vu}& 0\\ \mathrm{v}+\mathrm{w}& {\mathrm{v}}^2+{\mathrm{w}}^2+\mathrm{vw}& 0\\ {\mathrm{w}}^2& {\mathrm{w}}^3& 1\end{array}\right|=0$

$Exapand the determinant$

$$\begin{gathered} \left(u+v\right)\left(v^2+w^2+vw\right)-\left(v+w\right)\left(u^2+v^2+uv\right)=0 \\ \Rightarrow \cancel{u{v}^2}+u{w}^2+\cancel{uvw}+\cancel{v^3}+v{w}^2+\cancel{v^{2}w}-v{u}^2-\cancel{v^3}-\cancel{u{v}^2}-u^{2}w-\cancel{v^{2}w}-\cancel{uvw}=0 \\ \Rightarrow uw\left(w-u\right)+vw\left(w-v\right)=0 \end{gathered}$$

$\Rightarrow \mathrm{uv}+\mathrm{vw}+\mathrm{wu}=0$