To rotate the whole plane about the origin by an angle \theta, we only need to know where \hat{\imath} and \hat{\jmath} go — and trigonometry tells us exactly. The vector \hat{\imath} swings to (\cos\theta, \sin\theta), and \hat{\jmath} swings to (-\sin\theta, \cos\theta). Stacking those as columns gives the rotation matrix:

Turn the dial

Spin the angle below. The grid rotates rigidly — distances and angles are perfectly preserved, nothing stretches. At \theta = 90^\circ the matrix becomes \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, the quarter-turn that sends \hat{\imath} straight up.

Rotations that don't distort

A rotation is a rigid motion: it preserves every length and every angle, so it never changes area — its determinant is exactly 1. Its columns are perpendicular unit vectors, which makes it an orthogonal matrix, and undoing a rotation is simply rotating back: R(\theta)^{-1} = R(-\theta). Rotations are the workhorses of computer graphics, robotics and any place a thing needs to turn without warping.