where a y b and aab are the dot and cross products, respectively. One can easily verify that this multiplication is associative (multiplication of three or more quaternions is meaningful without parentheses), but it is clearly noncommutative (changing the order of quaternions in a product will, in general, yield a different answer). Taking a quaternion’s conjugate changes the sign of its vector part, thus: AA⊕ a a AA⊕ H-aL. Based on the corresponding Taylor expansion, we can easily construct functions of quaternion arguments, such as, for example,