4 entries of a rotation matrix with 2 rows and 2 columns. In 3-dimensional space, every rotation can be interpreted as a rotation by a given angle about a single fixed axis of rotation (see Euler's rotation theorem), and hence it can be simply described by an angle and a vector with 3 entries. However, it can also be represented by the 9 entries of a rotation matrix with 3 rows and 3 columns. The notion of rotation is not commonly used in dimensions higher than 3; there is a notion of a rotational displacement, which can be represented by a matrix, but no associated single axis or angle. Rotation matrices are square matrices, with real entries. More specifically they can be characterized as orthogonal matrices with determinant 1: . The set of all such matrices of size n forms a group, known as the special orthogonal group . In two dimensions In two dimensions every rotation matrix has the following form: . This rotates column vectors by means of the following matrix multiplication: . So the coordinates (x',y') of the point (x,y) after rotation are: , . The direction of vector rotation is counterclockwise if θ is positive (e.g. 90°), and clockwise if θ is negative (e.g. -90°). . Non-standard orientation of the coordinate system If a standard right-handed Cartesian coordinate system is used, with the x axis to the right and the y axis up, the rotation R(θ) is counterclockwise. If a left-handed Cartesian coordinate system is used, with x directed to the right but y directed down, R(θ) 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 y-axis 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° and 180° rotations: (90° counterclockwise rotation) (180° rotation in either direction – a half-turn) (270° counterclockwise rotation, the same as a 90° clockwise rotation) In Euclidean geometry, uniform scaling (isotropic scaling, homogeneous dilation, homothety) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions. The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is