The Rotation Group

The rotation group is the set of all rotations of a coordinate frame. One can write a realization of this group as a set of 3d matrices that map $\vr$ to $\vr'$ :

$$\left( \begin{array}{c} x' \\ y' \\ z' \end{array} \right) = ... ... \end{array} \right) \left( \begin{array}{c} x \\ y \\ z \end{array} \right)$$

This is tedious to write out! We will compress this notationally to this expression for each (the $i$ th) of the vector coordinates:

$$r_i' = \sum_{j=1}^3 R_{ij} r_j$$

In many physics books - especially at higher levels - it is pointless to even write the summation sign; people use the Einstein summation convention that repeated indices in an expression are to be summed:

$$r_i' = R_{ij} r_j$$

(three equations, note well, one each for $i = 1,2,3$ ).

One can easily prove that the transformations of this group leave the lengths (magnitudes) but not the directions of position vectors $\vert\vr\vert$ unchanged. Indeed, the ``correct'' way to derive a representation for the rotation matrix $R_{ij}$ (which we will also write $\Mat{R}$ ) is to find the set of all infinitesimal transformations that leave the length of a vector unchanged - their composition forms the Lie (continuous) rotation group, SO(3).