Principal Rotation Matrix

Rotation around an axis. Positive angle with the clock (Positive if vector with clock, and frame against clock).

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$$\begin{array}{rl}{\mathbf{R}}_z(\alpha )& =\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right]\\ {\mathbf{R}}_{\mathbf{y}}(\theta )& =\left[\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -\sin \theta & 0& \cos \theta \end{array}\right]\\ {\mathbf{R}}_x(\varphi )& =\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \varphi & -\sin \varphi \\ 0& \sin \varphi & \cos \varphi \end{array}\right]\end{array}$$