Relativistic Transformations. The Lorentz Group

A rotation1 may be specified by a vector, θ , in such a way that (Fig. 1.1.1) the rotation axis lies along θ , the rotation angle being θ = |θ|, and the direction of the rotation determined by the corkscrew rule. If we denote the rotation by R(θ ), it acts upon a vector r according to 1.1.1a$$r \to r = R\left( \theta \right)r = \left( {\cos \theta } \right)r + \left( {1 - \cos \theta } \right)\frac{{\theta r}}{{\theta ^2 }}\theta + \frac{{\sin \theta }}{\theta }\theta \times r;$$ for θ infinitesimal, 1.1.1b$$R\left( \theta \right)r = r + \theta \times r + O\left( {\theta ^2 } \right).$$