Rotations

Its columns are where $e_1$ and $e_2$ land after turning by $\theta$ anticlockwise: $e_1$ goes to $[\cos\theta, \sin\theta]$, and $e_2$ stays at 90° from it.

$\cos 90^\circ = 0$ and $\sin 90^\circ = 1$ give the quarter turn from before. For 30°, $\cos 30^\circ \approx 0.866$ and $\sin 30^\circ = 0.5$.

Same length ($\sqrt{5}$ both), new direction.

Turn two vectors by the same angle and the angle between them stays the same, so their dot product does too. In symbols: $(Ra)^\top (Rb) = a^\top R^\top R\, b = a^\top b$. A matrix with $R^\top R = I$ is called orthogonal.

Turning by $\beta$ and then by $\alpha$ is one turn by $\alpha + \beta$. And the transpose turns back the other way, so for rotations the inverse costs nothing: just transpose.

Multiply the matrices, then say which single rotation it is and where it sends $[1, 0]$.

$[1, 0]$ goes to $[-1, 0]$: every vector points the opposite way. The angles added: $90^\circ + 90^\circ = 180^\circ$.

Turn $q$ by $\alpha$ and $k$ by $\beta$. Their dot product depends only on $\beta - \alpha$, not on the two angles separately.

• $R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$.
• $R^\top R = I$: lengths and dot products are kept.
• $R(\alpha)R(\beta) = R(\alpha + \beta)$, $R(\theta)^{-1} = R(\theta)^\top = R(-\theta)$.
• $(R_\alpha q) \cdot (R_\beta k)$ depends only on $\beta - \alpha$: the idea behind RoPE.