a rotation of a point about the origin through an angle maps it to another point such that p and and p' are at the same distance from the origin and the angle from the vector p to the vector p' is .
to determine the coordinates of the point , it is very convenient to use polar coordinates. let , where is the distance from to the origin and the polar angle. then we have
we can write this as a linear map: we deduce following the transformation matrix: the matrix is called the rotation matrix. the inverse transform is the transpose which rotates vectors back through .
in projective coordinates the rotation matrix becomes: