If \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}\) and \(\overrightarrow{\mathbf{c}}\) are unit vectors, such that \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}}\), then \(3 \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}+2 \overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}\) is

\(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}\) are unit vectors, then \(|\overrightarrow{\mathbf{a}}|=|\overrightarrow{\mathbf{b}}|=|\overrightarrow{\mathbf{c}}|=1\)
Given,
\[
\begin{aligned}
\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}} &=0 \\
\overrightarrow{\mathbf{a}} &=-(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}})
\end{aligned}
\]
Squaring on both sides
\[
\begin{aligned}
\overrightarrow{\mathbf{a}^{2}} &=(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}})^{2} \\
\overrightarrow{\mathbf{a}^{2}} &=\overrightarrow{\mathbf{b}^{2}}+\overrightarrow{\mathbf{c}^{2}}+2 \overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}} \\
|\overrightarrow{\mathbf{a}}|^{2} &=|\overrightarrow{\mathbf{b}}|^{2}+|\overrightarrow{\mathbf{c}}|^{2}+2(\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}) \quad\left(\because \overrightarrow{\mathbf{a}^{2}}=|\overrightarrow{\mathbf{a}}|^{2}\right)
\end{aligned}
\]
\[
\begin{aligned}
&1=1+1+2(\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}})\\
&\Rightarrow \overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}=-1 / 2\\
&\text { Similarly, } \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}=-1 / 2\\
&\text { Hence, } 3 \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}+2 \overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}\\
&=3(-1 / 2)+2(-1 / 2)+(-1 / 2)\\
&=(3+2+1)(-1 / 2)\\
&=6(-1 / 2)=-3
\end{aligned}
\]