# Why is a small angle considered a vector?

I have been told by my teacher that the angle of rotation, namely theta cannot be considered as a vector, which is self explanatory as it does not follow the laws of vector algebra.

But then he said that a very very small angle (limit) can be considered as a vector because it has negligible effect on the vector mathematics, namely that vector a + vector b = vector b + vector a.

He also demonstrated the fact by rotating a book, and showed that theta is not a vector, but since a very small change in the angle will not have an effect, the small angle is considered an angle.

Hence = d"theta"/dx = $$\omega$$ (angular vecocity, which we know is a vector)

I do not understand how an angle, however small can be considered as a vector. Because no matter how much small you rotate something, that small change will effect the result even though it is tiny.

Does he mean it is not a vector because it is a pseudo-vector?

vanhees71
The rotation of a vector $\vec{x}$ around an axis given by the unit vector $\vec{n}$ and angle $\varphi$ is given by
$$\vec{x}'=(\vec{n} \cdot \vec{x}) \vec{n}+(\vec{n} \times \vec{x}) \times \vec{n} \cos \varphi+\vec{n} \times \vec{x} \sin \varphi.$$