representation of the (general) orthogonal group . In two dimensions In two dimensions, the standard rotation matrix has the following form: This rotates column vectors by means of the following matrix multiplication, Thus, the new coordinates of a point after rotation are Examples For example, when the vector is rotated by an angle , its new coordinates are and when the vector is rotated by an angle , its new coordinates are Direction The direction of vector rotation is counterclockwise if is positive (e.g. 90°), and clockwise if is negative (e.g. −90°). Thus the clockwise rotation matrix is found as The two-dimensional case is the only non-trivial (i.e. not one-dimensional) case where the rotation matrices group is commutative, so that it does not matter in which order multiple rotations are performed. An alternative convention uses rotating axes, and the above matrices also represent a rotation of the axes clockwise through an angle . Non-standard orientation of the coordinate system If a standard right-handed Cartesian coordinate system is used, with the to the right and the up, the rotation is counterclockwise. If a left-handed Cartesian coordinate system is used, with directed to the right but directed down, is clockwise. Such non-standard orientations are rarely used in mathematics but are common in 2D computer graphics, which often have the origin in the top left corner and the down the screen or page. See below for other alternative conventions which may change the sense of the rotation produced by a rotation matrix. Common rotations Particularly useful are the matrices for 90°, 180°, and 270° counter-clockwise rotations. Relationship with complex plane Since the matrices of the shape form a ring isomorphic to the field of the complex numbers . Under this isomorphism, the rotation matrices correspond to circle of the unit complex numbers, the complex numbers of modulus . If one identifies with through the linear isomorphism the action of a matrix of the above form on vectors of corresponds to the multiplication by the complex number , and rotations correspond to multiplication by complex numbers of modulus . As every rotation matrix can be written the above correspondence associates such a matrix with the complex number (this last equality is Euler's formula). In three dimensions Basic rotations A basic rotation (also called elemental rotation) is a rotation about one of the axes of a coordinate system. The following three basic rotation matrices rotate