The conjugate or inverse of q is defined as q* = qA-iqi-jq2~kq3 (D-3)

The quantity, qA, is the real or scalar part of the quaternion and iqt + jq2 + kq3 is the imaginary or vector part.

A vector in three-dimensional space, U, having components {/,, U2, U3 is expressed in quaternion notation as a quaternion with a scalar part of zero,

If the vector q corresponds to the vector part of q (i.e., q= iqt + jq2+kq3), then an alternative representation of q is i=(?*q) (D-5)

Quaternion multiplication is performed in the same manner as the multiplication of complex numbers or algebraic polynomials, except that the order of operations must be taken into account because Eq. (D-2) is not commutative. As an example, consider the product of two quaternions

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