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Knowing the dot product of two vectors, how can I establish the angle between them?
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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 $\mathbf{u} \cdot \mathbf{v}=0$.
by Diamond (89,175 points)
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