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Quaternions are hypercomplex numbers

where


We define the conjugate

the dot product

and the norm


By associating quaternions with vectors in , the product of two quaternions , can be written as a matrix-vector product



From the structure of these matrices, we can show the dot product equivalence






We can embed points into as follows:


It can be shown that unit quaternions represent rotations. In particular, given a rotation axis where and an angle , we can construct


Then, this rotation can be computed as



Note that and represent the same rotation since both the rotation axis as the magnitude are inverted.

It turns out that