If $\vec{a}$, $\vec{b}$, and $\vec{c}$ are three unit vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, then the value of $\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$ is:
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If the sum of $n$ unit vectors is $\vec{0}$, the sum of their pairwise dot products is always $-\frac{n}{2}$. Here, $n=3$, so $-\frac{3}{2}$.