representation of rotations, is a compact way of representing rotations in three dimensions. In its most compact form, it consists of a vector, the product of a unit vector ω that is the axis of rotation with the (signed) angle of rotation θ, so that the magnitude of the overall rotation vector θω equals the (unsigned) rotation angle. The quaternion associated with the rotation is In geometric algebra the rotation is represented by a bivector. This can be seen in its relation to quaternions. Let Ω be a unit bivector in the plane of rotation, and let θ be the angle of rotation. Then the rotation bivector is Ωθ. The quaternion closely corresponds to the exponential of half of the bivector Ωθ. That is, the components of the quaternion correspond to the scalar and bivector parts of the following expression: The exponential can be defined in terms of its power series, and easily evaluated using the fact that Ω squared is −1. So rotations can be represented by bivectors. Just as quaternions are elements of the geometric algebra, they are related by the exponential map in that algebra. Rotors The bivector Ωθ generates a rotation through the exponential map. The even elements generated rotate a general vector in three dimensions in the same way as quaternions: As in two dimensions, the quantity e-Ωθ/2 is called a rotor and written R. The quantity eΩθ/2 is then R−1, and they generate rotations as This is identical to two dimensions, except here rotors are four-dimensional objects isomorphic to the quaternions. This can be generalised to all dimensions, with rotors, elements of the even subalgebra with unit magnitude, being generated by the exponential map from bivectors. They form a double cover over the rotation group, so the rotors R and −R represent the same rotation. Matrices Axial vectors The rotation vector is an example of an axial vector. Axial vectors, or pseudovectors, are vectors with the special feature that their coordinates undergo a sign change relative to the usual vectors (also called "polar vectors") under inversion through the origin, reflection in a plane, or other orientation-reversing linear transformation. Examples include quantities like torque, angular momentum and vector magnetic fields. Quantities that would use axial vectors in vector algebra are properly represented by bivectors in geometric algebra. More precisely, if an underlying orientation is chosen, the axial vectors are naturally identified with the usual vectors;