Rotation Matrix

When discussing a Rotation, there are two possible conventions: rotation of the axes and rotation of the object relative to fixed axes.

In $\Bbb{R}^2$, let a curve be rotated by a clockwise Angle $\theta$, so that the original axes of the curve are $\hat{\bf x}$ and $\hat{\bf y}$, and the new axes of the curve are $\hat{\bf x}'$ and $\hat{\bf y}'$. The Matrix transforming the original curve to the rotated curve, referred to the original $\hat{\bf x}$ and $\hat{\bf y}$ axes, is

$${\hbox{\sf R}}_\theta = \left[{\matrix{\cos\theta & \sin\theta\cr -\sin\theta & \cos\theta\cr}}\right],$$ (1)

i.e.,

$${\bf x}={\hbox{\sf R}}_\theta{\bf x}'.$$ (2)

On the other hand, let the axes with respect to which a curve is measured be rotated by a clockwise Angle $\theta$, so that the original axes are $\hat{\bf x}_0$ and $\hat{\bf y}_0$, and the new axes are $\hat{\bf x}$ and $\hat{\bf y}$. Then the Matrix transforming the coordinates of the curve with respect to $\hat{\bf x}$ and $\hat{\bf y}$ is given by the Matrix Transpose of the above matrix:

$${\hbox{\sf R}}'_\theta = \left[{\matrix{\cos\theta & -\sin\theta\cr \sin\theta & \cos\theta\cr}}\right],$$ (3)

i.e.,

$${\bf x}={\hbox{\sf R}}'_\theta{\bf x}_0.$$ (4)

In $\Bbb{R}^3$, rotations of the $x$-, $y$-, and $z$-axes give the matrices

$${\hbox{\sf R}}_x(\alpha) = \left[\begin{array}{ccc}1 & 0 & 0\\ 0 & \cos\alpha & \sin\alpha\\ 0 & -\sin\alpha & \cos\alpha\end{array}\right]$$ (5)

$${\hbox{\sf R}}_y(\beta) = \left[\begin{array}{ccc}\cos\beta & 0 & -\sin\beta\\ 0 & 1 & 0\\ \sin\beta & 0 & \cos\beta\end{array}\right]$$ (6)

$${\hbox{\sf R}}_z(\gamma) = \left[\begin{array}{ccc}\cos\gamma & \sin\gamma & 0\\ -\sin\gamma & \cos\gamma & 0\\ 0 & 0 & 1\end{array}\right].$$ (7)

See also Euler Angles, Euler's Rotation Theorem, Rotation