If \(\vec{a},\vec{b}\) and \(\vec{c}\) are three unit vectors such that \(\vec{a}+\vec{b}+\vec{c}=0\) then the value of \(\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}\) is

\(-\frac{3}{2}\) \(\frac{3}{2}\) \(-1\) \(0\)

\(-\frac{3}{2}\)

\(\begin{aligned}\left(\vec{a}+\vec{b}+\vec{c}\right)^{2}&=0\\ |\vec{a}|^{2}+|\vec{b}|^{2}+|\vec{c}|^{2}+2(\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a})&=0\\ 3+2(\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a})&=0\\\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}&=-\frac{3}{2}\end{aligned} \)