 The quaternion product AC can now be represented as the matrix product

The product AC can also be viewed as an operation by C on the components of A, in which case we have

The dual quaternion product ÂĈ = (A, B)(C, D) = (AC, AD+BC) can be formulated as a matrix operation as follows.Assemble the components of Ĉ into the eight dimensional array Ĉ = (C1, C2, C3, c0, D1, D2, D3, d0), then ÂĈ is given by the 8x8 matrix product

As we saw for quaternions, the product ÂĈ can be viewed as the operation of Ĉ on the coordinate vector Â, which means ÂĈ can also be formulated as,

The dual quaternion of a displacement D=([A], d) can be constructed from the quaternion S=cos(φ/2) + sin(φ/2)S that defines the rotation [A] and the vector quaternion constructed from the translation vector d, given by D = d1i + d2j + d3k.Using this notation, the dual quaternion for the displacement D=([A], d) is given by

Let the Plücker coordinates of a line in the direction x through a point p in a moving body and its coordinates in the fixed frame which is in the direction X through the point P be given by,
Then the dual quaternion of the displacement of this body transforms Plücker coordinates in the moving frame to Plücker coordinates in the fixed frame by the formula

Using the matrix form of the dual quaternion product this becomes,
This calculation is easily managed using matrix operations.