A rotation matrix rotates a point (or vector) by a given angle around the origin. For a 3D
rotation, there are three degrees of freedom, so a different matrix is needed for a rotation
around each axis. But, for a 2D rotation, the only rotation which leaves the points in the
original plane is a rotation about the (imagined) Z axis.
2D Rotation
$$
R{\bf v} =
\begin{bmatrix}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta\\
\end{bmatrix}
\begin{bmatrix}
x\\
y\\
\end{bmatrix}
=
\begin{bmatrix}
x\cos\theta - y\sin\theta\\
x\sin\theta + y\cos\theta\\
\end{bmatrix}
$$
3D Rotation
$$
{\bf v} =
\begin{bmatrix}
x\\
y\\
z
\end{bmatrix}
$$
$$
R_x(\theta)=
\begin{bmatrix}
1 & 0 & 0\\
0 & \cos\theta & -\sin\theta\\
0 & \sin\theta & \cos\theta\\
\end{bmatrix}
$$
$$
R_y(\theta)=
\begin{bmatrix}
\cos\theta & 0 & \sin\theta\\
0 & 1 & 0\\
-\sin\theta & 0 & \cos\theta\\
\end{bmatrix}
$$
$$
R_z(\theta)=
\begin{bmatrix}
\cos\theta & -\sin\theta & 0\\
\sin\theta & \cos\theta & 0\\
0 & 0 & 1\\
\end{bmatrix}
$$