# Quaternions

This page is a sub-page of our page on Geometric Numbers in Euclidean 3D-space.

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Related KMR-pages:

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Other relevant sources of information:

Quaternion (at Wikipedia)
Rotor (at Wikipedia)
Versor (at Wikipedia)

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The quaternions represented as the even subalgebra
of the Clifford algebra $\, C_l(e_1, e_2, e_3) \,$ over the real numbers $\, \mathbb {R}$
.

$\, - e_1, - e_2, - e_3 \,$

$\, \textcolor {red} {e_1 e_2, e_2 e_3, e_1 e_3} \,$

This diagram shows the 1-blades (unbroken black arrows) and the 2-blades (red, dotted, broken arrows) among the blades in the canonical basis for $\, C_l(e_1, e_2, e_3)$.
The negatives of the 1-blades are shown as dotted black arrows.

The 2-blades $\, \textcolor {red} {e_1 e_2} \textcolor {black} {,} \textcolor {red} {e_2 e_3} \textcolor {black} {,} \textcolor {red} {e_1 e_3} \,$
represent the directed area within the corresponding squares.

$\, e_1^2 = e_2^2 = e_3^2 = 1 \,$

$\, e_k e_i = - e_i e_k , k \neq i \,$

Hence we have for $\, k \neq i \,$ : $\, (e_i e_k)^2 = e_i e_k e_i e_k = - e_k e_i e_i e_k = - e_k e_k = -1$

Addition rule for the even subalgebra of $\, C_l(e_1, e_2, e_3) \,$:

$\, (\alpha \textcolor {red} 1 + \alpha_{12} \textcolor {red} {e_1 e_2} + \alpha_{23} \textcolor {red} {e_2 e_3} \, + \alpha_{13} \textcolor {red} {e_1 e_3}) \, +$
$\, + \, (\alpha' \textcolor {red} 1 +\alpha'_{12} \textcolor {red} {e_1 e_2} + \alpha'_{23} \textcolor {red} {e_2 e_3} + \alpha'_{13} \textcolor {red} {e_1 e_3}) \stackrel {\mathrm{def}}{=} \,$
$\stackrel {\mathrm{def}}{=} \, (\alpha + \alpha') \textcolor {red} 1 + (\alpha_{12} + \alpha'_{12}) \textcolor {red} {e_1 e_2} + (\alpha_{23} + \alpha'_{23}) \textcolor {red} {e_2 e_3} + (\alpha_{13} + \alpha'_{13}) \textcolor {red} {e_1 e_3}$.

Multiplication table for the even subalgebra of $\, C_l(e_1, e_2, e_3) \,$:

$\, \begin{matrix} * & ~ & \textcolor {red} 1 & \textcolor {red} {e_1 e_2} & \textcolor {red} {e_2 e_3} & \textcolor {red} {e_1 e_3} \\ & & & & & & \\ \textcolor {red} 1 & ~ & \textcolor {red} 1 & \textcolor {red} {e_1 e_2} & \textcolor {red} {e_2 e_3} & \textcolor {red} {e_1 e_3} \\ \textcolor {red} {e_1 e_2} & ~ & \textcolor {red} {e_1 e_2} & - \textcolor {red} 1 & \textcolor {red} {e_1 e_3} & - \textcolor {red} {e_2 e_3} \\ \textcolor {red} {e_2 e_3} & ~ & \textcolor {red} {e_2 e_3} & - \textcolor {red} {e_1 e_3} & - \textcolor {red} 1 & \textcolor {red} {e_1 e_2} \\ \textcolor {red} {e_1 e_3} & ~ & \textcolor {red} {e_1 e_3} & \textcolor {red} {e_2 e_3} & - \textcolor {red} {e_1 e_2} & - \textcolor {red} 1 \, \end{matrix} \,$.

Multiplication table for the quaternions:

$\, \begin{matrix} * & ~ & \bold 1 & \;\; \bold i & \, \bold j & \, \bold k \\ & & & & & & \\ \bold 1 & ~ & \bold 1 & \;\; \bold i & \;\; \bold j & \;\; \bold k \\ \bold i & ~ & \bold i & - \bold 1 & \;\; \bold k & - \bold j \\ \bold j & ~ & \bold j & - \bold k & - \bold 1 & \;\; \bold i \\ \bold k & ~ & \bold k & \;\; \bold j & - \bold i & - \bold 1 \, \end{matrix} \,$.

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Substituting $\, \bold 1 = \textcolor {red} {1} \, , \, \bold i = \textcolor {red} {e_1 e_2} \, , \, \bold j = \textcolor {red} {e_2 e_3} \, , \bold k = \textcolor {red} {e_1 e_3} \,$ and comparing the two multiplication tables, we see that they are identical, and therefore they represent the same mathematical structure.

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What are quaternions, and how do you visualize them? A story of four dimensions.