Rotations

A rotation is a linear transformation that preserves lengths, angles, and orientation. In $\mathbb{R}^2$, rotation by angle $\theta$ counterclockwise is the matrix $R_\theta = \begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}$. Its columns are unit vectors 90° apart; it satisfies $R^\top R = I$ and $\det R = +1$ (the $+1$ is what distinguishes rotations from reflections, which have $\det = -1$).

Rotations in $\mathbb{R}^2$ form a group under composition: $R_\alpha R_\beta = R_{\alpha + \beta}$, rotations commute, and every rotation has an inverse $R_\theta^{-1} = R_{-\theta} = R_\theta^\top$. This beautiful structure is called $SO(2)$, the special orthogonal group in 2 dimensions.

In 3D, rotations are specified by an axis and an angle. Elementary rotations about the $x$-, $y$-, and $z$-axes have well-known $3\times 3$ matrix forms; any rotation can be decomposed into three such elementary ones via Euler angles. Unlike 2D, 3D rotations do not commute — rotating yaw-then-pitch is different from pitch-then-yaw, a fact pilots and robotics engineers know intimately.

Higher-dimensional rotations live in $SO(n)$, the group of orthogonal matrices with determinant $+1$. These preserve Euclidean distance and orientation in any number of dimensions. An important fact: every rotation can be written as $R = e^{A}$ for some skew-symmetric matrix $A$ (i.e., $A^\top = -A$). This is the bridge between the Lie group $SO(n)$ and its Lie algebra $\mathfrak{so}(n)$, used in robotics and differential geometry.

Worked example — composition of rotations: $R_{30°}R_{60°} = R_{90°}$ — rotations compose by *adding* angles. Algebraically, $\cos(30° + 60°) = \cos 30° \cos 60° - \sin 30° \sin 60°$, the well-known trig identity that emerges naturally from matrix multiplication.

Rotations matter in ML beyond graphics. In PCA, a data rotation aligns axes with directions of maximum variance. In normalizing flows, each invertible transformation is often a composition of rotations and scalings. Rotary Position Embedding (RoPE) in modern transformers encodes positions by rotating query/key vectors in high-dimensional space — a mathematically elegant way to inject sequence position that preserves inner products up to phase. Knowing your rotations is surprisingly practical.