Let $\overrightarrow{u}, \overrightarrow{v}\text{ and }\overrightarrow{w}$ be vectors such that $\overrightarrow{u}+\overrightarrow{v}+\overrightarrow{w}=0$. If $\left|\overrightarrow{u}\right|=3,\left|\overrightarrow{v}\right|=4\text{ and }\left|\overrightarrow{w}\right|=5$, then $\overrightarrow{u}.\overrightarrow{v}+\overrightarrow{v}\cdot \overrightarrow{w}+\overrightarrow{w}\cdot \overrightarrow{u}$ is 

Since $\overrightarrow{u}+\overrightarrow{v}+\overrightarrow{w}=0,\text{ we have }|\overrightarrow{u}+\overrightarrow{v}+\overrightarrow{w}{|}^2=0$

$\text{ or }|\overrightarrow{u}{|}^2+|\overrightarrow{v}{|}^2+|\overrightarrow{w}{|}^2+2(\overrightarrow{u}\cdot \overrightarrow{v}+\overrightarrow{v}\cdot \overrightarrow{w}+\overrightarrow{w}\cdot \overrightarrow{u})=0$

$\text{ or }9+16+25+2(\overrightarrow{u}\cdot \overrightarrow{v}+\overrightarrow{v}\cdot \overrightarrow{w}+\overrightarrow{w}\cdot \overrightarrow{u})=0$

$\text{ or }\overrightarrow{u}\cdot \overrightarrow{v}+\overrightarrow{v}\cdot \overrightarrow{w}+\overrightarrow{w}\cdot \overrightarrow{u}=-25$