Right-Handed System (Standard)

Thumb (X), Index (Y), Middle Finger (Z) point in +ve directions.
Rotation:
Counter-clockwise rotations are positive (by convention).

used by: OpenGL, in standart MATH.

\vec{X} \times \vec{Y} = \vec{Z}

Left-Handed System

Thumb (X), Index (Y), Middle Finger (Z) point in +ve directions.
If you curl fingers from X to Y, your thumb points along −Z (opposite of right-handed).
Clockwise rotations are positive (opposite convention).

\vec{X} \times \vec{Y} = - \vec{Z}

2D

[\begin{matrix} x^{'} \\ y^{'} \end{matrix}] = [\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]

3D

All rotation matrices are orthogonal: $R^{- 1} = R^{T}$ .
Determinant = +1: $det (R) = 1$ (preserves orientation).
Non-commutative: $R_{x} R_{y} \neq R_{y} R_{x}$ (order matters!).

Rotation Around X axis

R_{x} (θ) = [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos θ & - \sin θ \\ 0 & \sin θ & \cos θ \end{matrix}]

Rotates points in the YZ-plane (around the X-axis).
Leaves X-coordinates unchanged.

Rotation around Y-axis

R_{y} (θ) = [\begin{matrix} \cos θ & 0 & \sin θ \\ 0 & 1 & 0 \\ - \sin θ & 0 & \cos θ \end{matrix}]

Rotates points in the XZ-plane (around the Y-axis).
Leaves Y-coordinates unchanged.

Rotation around Z-axis

R_{z} (θ) = [\begin{matrix} \cos θ & - \sin θ & 0 \\ \sin θ & \cos θ & 0 \\ 0 & 0 & 1 \end{matrix}]

Rotates points in the XY-plane (around the Z-axis).
Leaves Z-coordinates unchanged.