Rotation Matrix

A 3×3 orthogonal matrix that represents a 3D rotation; multiplying it by a vector rotates that vector.

The rotation matrix is the most basic way to describe 3D orientation. The three columns of the 3×3 matrix R are the directions — as unit vectors, in the reference frame — of the object's own x, y, and z axes; each entry equals the cosine of the angle between two axes, which is why it's also called the direction cosine matrix. It must satisfy RᵀR = I (the columns are mutually perpendicular and unit length) and det R = 1; the set of all matrices meeting this condition forms the special orthogonal group SO(3). Of its 9 numbers, only 3 are independent — there's redundancy — but unlike Euler angles it has no singular points, and the math is simple: rotating a vector is just R times that vector, chaining two rotations is just matrix multiplication (order matters), and inverting a rotation is just a transpose. A rotation matrix plus a translation gives the 4×4 homogeneous transformation matrix.

ExampleThe rotation matrix for a rotation of θ about the z-axis is [[cosθ, −sinθ, 0], [sinθ, cosθ, 0], [0, 0, 1]]; at θ = 90°, the vector (1, 0, 0), originally pointing along x, rotates to (0, 1, 0).

Also called

Direction Cosine Matrix (DCM)

Related

Special Orthogonal Group SO(3) · Homogeneous Transformation Matrix · Quaternion · Axis-Angle Representation · 6D Rotation Representation · Coordinate Transformation

Sources

Wikipedia: Rotation matrix
Modern Robotics (Lynch & Park, 2017), 3.2.1 Rotation Matrices