$\vec{a} + \vec{b} + \vec{c} = \vec{0}$, with $\lvert \vec{a} \rvert = 2$, $\lvert \vec{b} \rvert = 3$ and $\lvert \vec{c} \rvert = 4$. What is $\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$?

Explanation

1. 01

   Square $\vec{a} + \vec{b} + \vec{c} = \vec{0}$: $\lvert \vec{a} \rvert^{2} + \lvert \vec{b} \rvert^{2} + \lvert \vec{c} \rvert^{2} + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0$.

2. 02

   $4 + 9 + 16 = 29$, so the sum is $-\frac{29}{2}$.