If θ is the angle between a and b, what is tan θ?

Explanation

1. 01

   $\tan \theta = \frac{\lvert \vec{a} \times \vec{b} \rvert}{\vec{a} \cdot \vec{b}}$, since $\lvert \vec{a} \rvert \lvert \vec{b} \rvert$ cancels from $\sin \theta$ and $\cos \theta$.

2. 02

   $\vec{a} \cdot \vec{b} = 4 + 3 - 4 = 3$, so $\tan \theta = \frac{12}{3} = 4$.