## Q qqm qq q2q2 q3q3 qrfj 23

+j( ~ lifi + + + Ml) + k(qlq'2-q2q\ + q3q^+q^q3) (D-7)

If q' = (q\,q'), then Eq. (D-7) can alternatively be expressed as q"=-qq' = (q^ - q• q', ?4q'+ q'A + qX q) The length or norm of q is defined as

If a set of four Euler symmetric parameters corresponding to the rigid body rotation defined by the transformation matrix, A (Section 12.1), are the components of the quaternion, q, then q is a representation of the rigid body rotation. If q' corresponds to the rotation matrix A', then the rotation described by the product A'A is equivalent to the rotation described by qqf. (Note the inverse order of quaternion multiplication as compared with matrix multiplication.)

The transformation of a vector U, corresponding to multiplication by the matrix A,

is effected in quaternion algebra by the operation

See Section 12.1 for additional properties of quaternions used to represent rigid., body rotations.

For computational purposes, it is convenient to express quaternion multiplication in matrix form. Specifically, let the components of q form a four-vector as follows:

This procedure is analogous to expressing the complex number c <= a+ ib in the form of the two-vector,

In matrix form, Eq. (D-7) then becomes

0 0