74. Rotation Matrix
A rotation matrix rotates vectors about the origin by an angle without changing their lengths.
74.1. 2D rotation
Rotation by angle counterclockwise about the origin:
Applied to :
Example
Rotate by ():
The unit -vector becomes the unit -vector. ✓
74.2. Properties
- Orthogonal: , so — see Orthogonal Matrix
- Determinant: — orientation-preserving
- Composition: — rotations add
- Inverse:
- Preserves lengths and angles
74.3. 3D rotations (about coordinate axes)
About the -axis:
About the -axis:
About the -axis:
74.4. General 3D rotation
Any 3D rotation is a composition of rotations about coordinate axes. Common parameterizations:
- Euler angles: — three angles
- Axis–angle (Rodrigues formula): rotate by angle about unit axis :
where is the skew-symmetric “cross-product matrix” of
- Quaternions: — avoids gimbal lock, used in 3D graphics / robotics
74.5. Eigenvalues
For a 2D rotation by :
Real eigenvalues only when () or (). Otherwise the eigenvalues are complex — rotations have no real eigenvectors (no fixed direction in the plane).
74.6. Connection to special orthogonal group
The set of all rotation matrices forms the special orthogonal group :
Excluding the determinant case removes reflections — see Reflection Matrix.