If ABC is an equilateral triangle of side a, then the value of $\vec{AB}. \vec{BC} +\vec{BC}.\vec{CA} +\vec{CA}.\vec{AB}$ is equal to

$-\frac{3a^2}{2}$

We have,

$\vec{AB}+\vec{BC}+\vec{CA}=\vec 0$

$⇒|\vec{AB}+\vec{BC}+\vec{CA}|^2=0$

$⇒|\vec{AB}|^2+|\vec{BC}|^2+|\vec{CA}|^2+2(\vec{AB}. \vec{BC} +\vec{BC}.\vec{CA}+\vec{CA}.\vec{AB})=0$

$⇒\vec{AB}. \vec{BC} +\vec{BC}.\vec{CA}+\vec{CA}.\vec{AB}=-\frac{3a^2}{2}$