Rotations in two dimensions are easily described; just specify the angle of rotation and the orientation (clockwise or counterclockwise). In three dimensions, it is also necessary to specify the axis of rotation. By convention, a positive angle of rotation about that axis implies rotation in a counterclockwise direction looking back along the axis. Thus, a rotation by $\frac\pi2$ about the North Pole corresponds to spinning the globe $\frac14$ of the way around to the east; spinning to the west would correspond either to a negative rotation about the North Pole, or to a positive rotation about the South Pole.

How do you specify a rotation in higher dimensions? In three dimensions, specifying the axis of rotation is just a way of specifying the plane in which the rotation takes place. In higher dimensions, one must specify the plane of rotation itself, as there is more than one “axis” perpendicular to any plane.

A key property of rotations is that they preserve length! Rotations take a sphere to itself, and hence take any vector to some other vector of the same length. In other words, the norm $|\vv|$ of a vector is unchanged by a rotation. However, rotations are not the only transformations with this property; there are also reflections. The difference is that rotations are orientation-preserving, whereas reflections are orientation-reversing. (These are the only possibilities; any linear transformation must result in either an even or an odd permutation of the relative positions of the axes.)

So in higher dimensions, we define a rotation to be a length-preserving, orientation-preserving linear transformation of a real vector space. If you preserve length and orientation, and then do so again, you have clearly preserved length and orientation, so the composition of two rotations is again a rotation. Furthermore, you can undo any (sequence of) rotation(s) simply by performing the same rotation(s) in the opposite direction (and in the opposite order). These two properties — closure under composition and the existence of an inverse — ensure that the collection of all rotations is a group. 1)

The group of length-preserving linear transformations in $n$ dimensions is called $\Og(n)$, the orthogonal group in $n$ dimensions; those transformations which are also orientation-preserving make up the rotation group in $n$ dimensions, $\SO(n)$.

These groups are normally expressed in terms of matrices. Since we are studying transformations of $\RR^n$, these matrices are real, so that matrix multiplication is associative. Consider first linear transformations in two dimensions, which take a vector \begin{equation} v = \begin{pmatrix} x\\ y\\ \end{pmatrix} \end{equation} to another vector \begin{equation} w = M v \end{equation} where $M$ is some matrix \begin{equation} M = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \end{equation} where $a,b,c,d,x,y\in\RR$. The (squared) length of $v$ is given by \begin{equation} |v|^2 = x^2 + y^2 = v^T v \end{equation} where $v^T$ denotes the matrix transpose of $v$. Thus, the orthogonal group $\Og(2)$ consists of those matrices $M$ that preserve the length of $v$, that is, for which \begin{equation} v^T v = (Mv)^T (Mv) = v^T M^T M v \end{equation} for any $v\in\RR^2$. This can only be true if \begin{equation} M^T M = I \end{equation} where $I$ is the ($2\times2$) identity matrix. Since \begin{equation} \det(M^T) = \det(M) \end{equation} we must have \begin{equation} \det M = \pm1 \end{equation} and it is straightforward to generalize this construction to higher dimensions. The rotation groups can therefore be expressed as \begin{align} \Og(n) &= \{ M\in\RR^{n\times n} : M^T M = I \} \\ \SO(n) &= \{ M\in\RR^{n\times n} : M^T M = I, \det M = 1 \} \end{align} where we have introduced the notation $\KK^{m\times n}$ for the $m\times n$ matrices whose elements lie in $\KK$; these expressions are often taken as the definitions of the orthogonal and special orthogonal groups, respectively.

In the remainder of this chapter, we set the stage for later developments by discussing the properties of several important orthogonal groups. In certain dimensions, orthogonal transformations can also be expressed in terms of division algebras other than $\RR$; we save that discussion for §7.2 and §9.1.