# Why is the angle between vectors restricted?

Why is the calculated angle between two vectors always between $$\pi$$ and $$0$$.

Is this due to the limitations of $$\arccos\theta$$ or is it because angles between vectors is described to be the smaller one?

## 2 Answers

It's a matter of tradition and simplicity.

Each pair of vectors defines two angles. One of them is between $$0$$ and $$\pi$$ and the other is between $$\pi$$ and $$2\pi$$. Since their sum is $$2\pi$$, we only really need to tell one of the two angles, and we decide, for simplicity, to always report the smaller of the two.

• It might be worth additionally pointing out that in certain (?) settings, like the plane, we can make a canonical choice of the "direction of the angle", and thus report the angle from a vector to another to always be the one in positive (counter-clockwise) direction. In more general settings such a canonical direction is not available. – Mees de Vries Apr 23 at 11:49

There are angles of all possible degrees. In context with angles in a triangle however, we have $$0\le \theta \le \pi$$.