Question

Knowing the dot product of two vectors, how can I establish the angle between them?

Answer 1

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 $\mathbf{u} \cdot \mathbf{v}=0$.