 The rotation part can be given by

where formula_37 is the angle of rotation about the direction given by unit vector formula_38.The displacement part can be written as

The dual-quaternion equivalent of a 3D-vector is

and its transformation by formula_41 is given by

These dual quaternions (or actually their transformations on 3D-vectors) can be represented by the homogeneous transformation matrix
where the 3×3 orthogonal matrix is given by

For the 3D-vector

the transformation by T is given by

Besides being the tensor product of two Clifford algebras, the quaternions and the dual numbers, the dual quaternions have two other formulations in terms of Clifford algebras.First, dual quaternions are isomorphic to the Clifford algebra generated by 3 anticommuting elements i, j, e with i2 = j2 = -1 and e2 = 0.