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The examples in this article apply to active rotations of vectors counterclockwise in a right-handed coordinate system (y counterclockwise from x) by pre-multiplication (the rotation matrix R applied on the left of the column vector v to be rotated).
Rotation matrices have a determinant of +1, and reflection matrices have a determinant of −1. The set of all orthogonal two-dimensional matrices together with matrix multiplication form the orthogonal group: O(2). The following table gives examples of rotation and reflection matrix :
By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra so(3) to its Lie group SO(3).
Bivectors are related to the eigenvalues of a rotation matrix. Given a rotation matrix M the eigenvalues can be calculated by solving the characteristic equation for that matrix 0 = det(M − λI). By the fundamental theorem of algebra this has three roots (only one of which is real as there is only one eigenvector, i.e., the axis of rotation ...
That it is an orthogonal matrix means that its rows are a set of orthogonal unit vectors (so they are an orthonormal basis) as are its columns, making it simple to spot and check if a matrix is a valid rotation matrix. Above-mentioned Euler angles and axis–angle representations can be easily converted to a rotation matrix.
A rotation of the vector through an angle θ in counterclockwise direction is given by the rotation matrix: = ( ), which can be viewed either as an active transformation or a passive transformation (where the above matrix will be inverted), as described below.
Similarly, a rotation matrix requires orthogonal basis vectors, so in 3D space the third vector can unambiguously be calculated from the first two vectors with a cross product (though there is ambiguity in the sign of the third vector if improper rotations are allowed).
The geometric interpretation of curl as rotation corresponds to identifying bivectors (2-vectors) in 3 dimensions with the special orthogonal Lie algebra of infinitesimal rotations (in coordinates, skew-symmetric 3 × 3 matrices), while representing rotations by vectors corresponds to identifying 1-vectors (equivalently, 2-vectors) and ...