Using quaternions to describe finite rotations brings attention to their capacity to specify arbitrary rotations in space without degeneration to singularity and to their usefulness in extending the vector algebra to encompass multiplication and division for both scalars and spatial vectors. A quaternion contains four parameters, and they have been proved to be a minimal set for defining a nonsingular mapping between the parameters and their corresponding rotational transformation matrix. Many useful identifies pertaining to quaternion multiplications are generalized in this paper. Among them multiplicative commutativity is the most powerful. Since quaternion space includes the three-dimensional vector space, the physical quantities related to rotations, such as angular displacement, velocity, acceleration, and momentum, are shown to be vector quaternions, and their expressions in quaternion space are derived. These kinematic and dynamic differential equations are further shown to be invertible due to the fact that they are written in quaternion space, and the highest order term of the rotation parameters can be expressed explicitly in closed form.
