Step-by-step Solution:

\[ \text{Here, } \vec{a} + \vec{b} + \vec{c} = 0 \text{ and } \vec{a}^2 = 4, \quad \vec{b}^2 = 9, \quad \vec{c}^2 = 25 \] \[ \therefore (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c}) = 0 \] \[ \Rightarrow \vec{a}^2 + \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} + \vec{b} \cdot \vec{a} + \vec{b}^2 + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} + \vec{c}^2 = 0 \] \[ \Rightarrow \vec{a}^2 + \vec{b}^2 + \vec{c}^2 + 2 (\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \quad [\because \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}] \] \[ \Rightarrow 4 + 9 + 25 + 2 (\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \] \[ \Rightarrow \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = \frac{-38}{2} = -19 \]