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Topic B2
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Consider these important definitions that apply to all following explanations.

The quaternion set is given by

:math:`\mathbb{H}\triangleq\left\{ h_{1}+\imi h_{2}+\imj h_{3}+\imk h_{4}\,:\,h_{1},h_{2},h_{3},h_{4}\in\mathbb{R}\right\}`

in which the imaginary units :math:`\imi`, :math:`\imj`, and :math:`\imk` have the following properties:

:math:`\hat{\imath}^{2}=\hat{\jmath}^{2}=\hat{k}^{2}=\hat{\imath}\hat{\jmath}\hat{k}=-1`

The dual quaternion set is given by

:math:`\mathcal{H}\triangleq\left\{ \quat h+\dual\quat h'\,:\,\quat h,\quat h'\in\mathbb{H},\,\dual^{2}=0,\,\dual\neq0\right\}`

where :math:`\dual^2=0` but :math:`\dual\neq0`.