page_title: Rotation matrix
section_title: Non-standard orientation of the coordinate system
hierarchical_section_title: Rotation matrix / In two dimensions / Non-standard orientation of the coordinate system
caption_attribution_description: English: This image has been created to illustrate "rotation matrix" article It shows a clockwise rotation of angle theta in axes that are *not* oriented in the standard way. x goes to the right and y goes to the bottom.
context_page_description: In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
  rotates points in the xy-plane counterclockwise through an angle θ with respect to the x axis about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v =, it should be written as a column vector, and multiplied by the matrix R:
  The examples in this article apply to active rotations of vectors counterclockwise in a right-handed coordinate system by pre-multiplication. If any one of these is changed, then the inverse of the example matrix should be used, which coincides with its transpose.
  Since matrix multiplication has no effect on the zero vector, rotation matrices describe rotations about the origin. Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. In some literature, the term rotation is generalized to include improper rotations, characterized by orthogonal matrices with determinant −1. These combine proper rotations with reflections.
context_section_description: 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.
caption: A rotation through angle θ with non-standard axes.
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