In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy-Cartesian plane counter-clockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation using a rotation matrix R, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv. Rotation matrices also provide a means of numerically representing an arbitrary rotation of the axes about the origin, without appealing to angular specification. These coordinate rotations are a natural way to express the orientation of a camera, or the attitude of a spacecraft, relative to a reference axes-set. Once an observational platform's local X-Y-Z axes are expressed numerically as three direction vectors in world coordinates, they together comprise the columns of the rotation matrix R (world → platform) that transforms directions (expressed in world coordinates) into equivalent directions expressed in platform-local coordinates. The examples in this article apply to active rotations of vectors counter-clockwise in a right-handed coordinate system by pre-multiplication. If any one of these is changed (e.g. rotating axes instead of vectors, i.e. a passive transformation), then the inverse of the example matrix should be used, which coincides precisely with its transpose. Since matrix multiplication has no effect on the zero vector (the coordinates of the origin), rotation matrices can only be used to describe rotations about the origin of the coordinate system. Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. Rotation matrices are square matrices, with real entries. More specifically, they can be characterized as orthogonal matrices with determinant 1; that is, a square matrix R is a rotation matrix if RT = R−1 and det R = 1. In some literature, the term rotation is generalized to include improper rotations, characterized by orthogonal matrices with determinant −1 (instead of +1). These combine proper rotations with reflections (which invert orientation). In other cases, where reflections are not being considered, the label proper may be dropped. This convention is followed in this article. The set of all orthogonal matrices of size n with determinant +1 forms a group known as the special orthogonal group SO(n). The most important special case is that of the rotation group SO(3). The set of all orthogonal matrices of size n with determinant +1 or -1 forms the (general) orthogonal group O(n).

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