$\vec{a} + \vec{b} + \vec{c} = \vec{0}$, with $\lvert \vec{a} \rvert = 3$, $\lvert \vec{b} \rvert = 8$ and $\lvert \vec{c} \rvert = 7$. What is the angle between $\vec{a}$ and $\vec{b}$?

Explanation

1. 01

   $\vec{c} = -(\vec{a} + \vec{b})$, so $\lvert \vec{c} \rvert^{2} = \lvert \vec{a} \rvert^{2} + 2\vec{a} \cdot \vec{b} + \lvert \vec{b} \rvert^{2}$: $49 = 9 + 2\vec{a} \cdot \vec{b} + 64$.

2. 02

   $\vec{a} \cdot \vec{b} = -12$, so $\cos \theta = \frac{-12}{3 \times 8} = -\frac{1}{2}$ and $\theta = \frac{2\pi}{3}$.