page_title: Rotation matrix
section_title: In two dimensions
hierarchical_section_title: Rotation matrix / In two dimensions
caption_attribution_description: English: This image has been created to illustrate "rotation matrix" page It shows a rotation of angle theta in a plane oriented using the usual conventions (x to the right and y to the top)
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: 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 (x′, y′) of a point (x, y) after rotation are
  .
caption: A counterclockwise rotation of a vector through angle  θ. The vector is initially aligned with the x-axis.
exif: {}