[NOTES/CM-08015] Rotations in Three Dimensions

Rotations

Consider two sets of orthogonal coordinate axes \(K\) and \(K\Prime\) with a common origin. Let \((x_1,x_2,x_3)\) denote the components of position vector of a point \(P\) w.r.t. the axes \(K\). Also let the components of the position vector of the same point w.r.t. \(K\Prime\) be \((x_1\Prime,x_2\Prime,x_3\Prime)\). Then the two set of components are linearly related and we write \begin{equation}
\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix} = R \begin{pmatrix}x_1\Prime\\x_2\Prime\\x_3\Prime\end{pmatrix} \end{equation}
where \(R\) is a \(3\times 3\) matrix. \\ We will say that the coordinate system \(K\Prime\) is obtained from \(K\) by a rotation represented mathematically be the matrix \(R\), to be called rotation matrix. 

Proper and Improper Rotations

 We list a few statements about rotations without details. 

1. The matrix \(R\) corresponding to a rotation is an orthogonal matrix.
2. An orthogonal matrix \(R\) can have \(\det R=\pm 1\). The rotations with \(\det R=+1\) .
3. An expression for the matrix \(R\) can be written down in terms of direction cosines of \(K\) w.r.t. \(K\Prime\).
4. The most important and frequently used rotations are the rotations which can be implemented physically. These rotations, do not change the handedness of the coordinate axes and are called proper rotations,
5. Two systems of coordinate axes related by matrix \(R\) have the same handedness if and only if \(\det R=1\).
6. A rotation with \(\det R=-1\) relates systems with opposite handedness. Such a rotation is called an improper rotation.

Euler's Theorem on rotations

Let any \(A\) and \(B\) be two possible configurations of a rigid body with one point fixed. Then it is possible to bring the body from the configuration \(A\) to the configuration \(B\) by simply rotating it about some definite line through the fixed point. In other words, a rotation about a point is always equivalent to a rotation about a line through the fixed point. 

Euler's theorem can be reformulated as follows.
Given two sets of right handed (or left handed) systems of axes, \(K\) and \(K\Prime\), with a common origin, we can always find a line passing through the origin such that by a rotation the \(K\) axes can be made to coincide with \(K\Prime\) axes. It also means that composition of two rotations is also a rotation.