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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 = [tex]\omega[/tex] (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.