perpendicular to . In this specific case, the relationship between the linear velocity vector and the angular velocity vector is (see circular motion and cross product). By the transitivity of the abovementioned equations, which implies Quaternion ↔ angular velocities The angular velocity vector can be obtained from the derivative of the quaternion as follows: where is the conjugate (inverse) of . Conversely, the derivative of the quaternion is Rotors in a geometric algebra The formalism of geometric algebra (GA) provides an extension and interpretation of the quaternion method. Central to GA is the geometric product of vectors, an extension of the traditional inner and cross products, given by where the symbol denotes the exterior product or wedge product. This product of vectors , and produces two terms: a scalar part from the inner product and a bivector part from the wedge product. This bivector describes the plane perpendicular to what the cross product of the vectors would return. Bivectors in GA have some unusual properties compared to vectors. Under the geometric product, bivectors have a negative square: the bivector describes the -plane. Its square is . Because the unit basis vectors are orthogonal to each other, the geometric product reduces to the antisymmetric outer product – and can be swapped freely at the cost of a factor of −1. The square reduces to since the basis vectors themselves square to +1. This result holds generally for all bivectors, and as a result the bivector plays a role similar to the imaginary unit. Geometric algebra uses bivectors in its analogue to the quaternion, the rotor, given by where is a unit bivector that describes the plane of rotation. Because squares to −1, the power series expansion of generates the trigonometric functions. The rotation formula that maps a vector to a rotated vector is then where is the reverse of (reversing the order of the vectors in is equivalent to changing its sign). Example. A rotation about the axis can be accomplished by converting to its dual bivector, where is the unit volume element, the only trivector (pseudoscalar) in three-dimensional space. The result is In three-dimensional space, however, it is often simpler to leave the expression for , using the fact that commutes with all objects in 3D and also squares to −1. A rotation of the vector in this plane by an angle is then Recognizing that and that is the