Rotation Matrix

74. Rotation Matrix

A rotation matrix rotates vectors about the origin by an angle $𝜃$ without changing their lengths.

74.1. 2D rotation

Rotation by angle $𝜃$ counterclockwise about the origin:

𝑅 (𝜃) = [\begin{matrix} cos 𝜃 & - sin 𝜃 \\ sin 𝜃 & cos 𝜃 \end{matrix}]

Applied to $\vec{𝑥} = [\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}]$ :

𝑅 (𝜃) \vec{𝑥} = [\begin{matrix} 𝑥 cos 𝜃 - 𝑦 sin 𝜃 \\ 𝑥 sin 𝜃 + 𝑦 cos 𝜃 \end{matrix}]

Example

Rotate $[\begin{matrix} 1 \\ 0 \end{matrix}]$ by $𝜃 = 90 °$ ( $\frac{𝜋}{2}$ ):

𝑅 (\frac{𝜋}{2}) = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}]

𝑅 (\frac{𝜋}{2}) [\begin{matrix} 1 \\ 0 \end{matrix}] = [\begin{matrix} 0 \\ 1 \end{matrix}]

The unit $𝑥$ -vector becomes the unit $𝑦$ -vector. ✓

74.2. Properties

Orthogonal: ${𝑅 (𝜃)}^{𝑇} 𝑅 (𝜃) = 𝐼$ , so $𝑅^{- 1} = 𝑅^{𝑇}$ — see Orthogonal Matrix
Determinant: $det (𝑅 (𝜃)) = {cos}^{2} 𝜃 + {sin}^{2} 𝜃 = 1$ — orientation-preserving
Composition: $𝑅 (𝛼) 𝑅 (𝛽) = 𝑅 (𝛼 + 𝛽)$ — rotations add
Inverse: ${𝑅 (𝜃)}^{- 1} = 𝑅 (- 𝜃)$
Preserves lengths and angles

74.3. 3D rotations (about coordinate axes)

About the $𝑥$ -axis:

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

About the $𝑦$ -axis:

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

About the $𝑧$ -axis:

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

74.4. General 3D rotation

Any 3D rotation is a composition of rotations about coordinate axes. Common parameterizations:

Euler angles: $𝑅_{𝑧} (𝛼) 𝑅_{𝑦} (𝛽) 𝑅_{𝑥} (𝛾)$ — three angles
Axis–angle (Rodrigues formula): rotate by angle $𝜃$ about unit axis $\hat{𝑘}$ :

𝑅 = 𝐼 + sin 𝜃 𝐾 + (1 - cos 𝜃) 𝐾^{2}

where $𝐾$ is the skew-symmetric “cross-product matrix” of $\hat{𝑘}$

Quaternions: $𝑅 = 𝑞 \cdot \vec{𝑥} \cdot 𝑞^{*}$ — avoids gimbal lock, used in 3D graphics / robotics

74.5. Eigenvalues

For a 2D rotation by $𝜃$ :

𝜆 = 𝑒^{\pm 𝑖 𝜃} = cos 𝜃 \pm 𝑖 sin 𝜃

Real eigenvalues only when $𝜃 = 0$ ( $𝜆 = 1$ ) or $𝜃 = 𝜋$ ( $𝜆 = - 1$ ). 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 $𝑛 \times 𝑛$ rotation matrices forms the special orthogonal group $𝑆 𝑂 (𝑛)$ :

𝑆 𝑂 (𝑛) = {𝑄 \in ℝ^{𝑛 \times 𝑛} : 𝑄^{𝑇} 𝑄 = 𝐼, det (𝑄) = 1}

Excluding the $- 1$ determinant case removes reflections — see Reflection Matrix.