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Knowing the dot product of two vectors, how can I establish the angle between them?
in Mathematics by Platinum (108k points) | 344 views

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Best answer

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 Platinum (108k points)
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