Why is a small angle considered a vector?

[AlchemistK]

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.

 

[kcdodd]

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

 

[vanhees71]

I don't understand the "explanations" of this teacher at all. A rotation can be described by giving the direction of the rotation axis and the rotation angle. Then the rotation is uniquely determined by the right-hand rule: Put the thumb in direction of the rotation axis. Then your fingers give the direction of rotation. Such vectors are known as axial vectors or pseudo vectors. They behave in any respect as usual polar vectors as long as you consider rotations, but they behave differently under space reflections. A polar vector changes sign under space reflections, while an axial vector doesn't change.

The rotation of a vector $\vec{x}$ around an axis given by the unit vector $\vec{n}$ and angle $\varphi$ is given by

$$\vec{x}'=(\vec{n} \cdot \vec{x}) \vec{n}+(\vec{n} \times \vec{x}) \times \vec{n} \cos \varphi+\vec{n} \times \vec{x} \sin \varphi.$$