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Quaternions are hypercomplex numbers

where
i^2 = j^2 = k^2 = ijk = -1

We define the conjugate
\bar{q} = a - ib - jc - kd
the dot product
\langle q, q' \rangle = aa' + ibb' + jcc' + kdd'
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 \| n \| = 1 and an angle , we can construct


Then, this rotation can be computed as

p' = M_R(\bar{q}) M_L(q) p

Note that and \bar{q} represent the same rotation since both the rotation axis as the magnitude are inverted.

It turns out that
M_R(\bar{q}) M_L(q) = \begin{pmatrix} 1 & 0 \\ 0 & R_{n, \theta} \end{pmatrix}