Rewriting the formula

$$\mathbf{u} \cdot \mathbf{v}=\|\mathbf{u}\|\|\mathbf{v}\| \cos \theta$$

we can rearrange and obtain

$$

\cos \theta=\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}

$$

Since $0 \leq \theta \leq \pi$, it follows and properties of the cosine function studied in trigonometry that

- $-\theta$ is acute if $\mathbf{u} \cdot \mathbf{v}>0$
- $ \theta$ is obtuse if $\mathbf{u} \cdot \mathbf{v}<0 $
- $\theta=\pi / 2$ if $\mathbf{u} \cdot \mathbf{v}=0$.