$\lvert \vec{a} \rvert = 4$, $\lvert \vec{b} \rvert = 3$, and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{2\pi}{3}$. What is $\vec{a} \cdot \vec{b}$?

Explanation

1. 01

   $\vec{a} \cdot \vec{b} = \lvert \vec{a} \rvert \lvert \vec{b} \rvert \cos \theta = 4 \times 3 \times \cos\frac{2\pi}{3}$.

2. 02

   $\cos\frac{2\pi}{3} = -\frac{1}{2}$, so $\vec{a} \cdot \vec{b} = -6$.