Given three vectors \(\bar{a}, \bar{b}, \bar{c}\), two of which are collinear. If \(\bar{a}+\bar{b}\) is collinear with \(\bar{c}\) and \(\bar{b}+\bar{c}\) is collinear with \(|\bar{a}|=|\bar{b}|=|\bar{c}|=\sqrt{2}\), then \(\bar{a} \cdot \bar{b}+\bar{b} \cdot \bar{c}+\bar{c} \cdot \bar{a}=\)

\(\vec{a}+\vec{b}=\lambda \vec{c} \quad \ldots(i) \text { and } \vec{b}+\vec{c}=\mu \vec{a} \quad \ldots(i i) \)
\( \Rightarrow \vec{a}-\vec{c}=\lambda \vec{c}-\mu \vec{a} \quad \text { form (i) and (ii) } \)
\( \Rightarrow(1+\mu) \vec{a}=(1+\lambda) \vec{c} \)
\( \Rightarrow \mu=\lambda=-1 \)
\( \Rightarrow \vec{a}+\vec{b}+\vec{c}=0 \)
\( \Rightarrow|\vec{a}+\vec{b}+\vec{c}|^2=|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}) \)
\( \Rightarrow 0=2+2+2+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}) \)
\( \Rightarrow \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}=-3\)