If three distinct points $\mathrm{P}\left(3{\mathrm{u}}^2, 2{\mathrm{u}}^3\right); \mathrm{Q}\left(3{\mathrm{v}}^2, 2{\mathrm{v}}^3\right)$ and $\mathrm{R}\left(3{\mathrm{w}}^2, 2{\mathrm{w}}^3\right)$ are collinear, then uv + vw + wu is equal to _________.

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

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

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

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

${\mathrm{R}}_1\to {\mathrm{R}}_1-{\mathrm{R}}_2$

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

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

or $\left({\mathrm{v}}^2+{\mathrm{w}}^2+\mathrm{vw}\right)-(\mathrm{v}+\mathrm{w})\left[\right(\mathrm{v}+\mathrm{w})+\mathrm{u}]=0$

or ${\mathrm{v}}^2+{\mathrm{w}}^2+\mathrm{vw}-(\mathrm{v}+\mathrm{w}{)}^2-\mathrm{u}(\mathrm{v}+\mathrm{w})=0$

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