Paravector - The Three-dimensional Euclidean Space

The Three-dimensional Euclidean Space

The following list represents an instance of a complete basis for the space,

which forms an eight-dimensional space, where the multiple indices indicate the product of the respective basis vectors, for example

The grade of a basis element is defined in terms of the vector multiplicity, such that

Grade	Type	Basis element/s
0	Unitary real scalar
1	Vector
2	Bivector
3	Trivector volume element

According to the fundamental axiom, two different basis vectors anticommute,

$\mathbf{e}_i \mathbf{e}_j + \mathbf{e}_j \mathbf{e}_i = 2 \delta_{ij}$

or in other words,

$\mathbf{e}_i \mathbf{e}_j = - \mathbf{e}_j \mathbf{e}_i \,\,; i \neq j$

This means that the volume element squares to

$\mathbf{e}_{123}^2 = \mathbf{e}_1 \mathbf{e}_2 \mathbf{e}_3 \mathbf{e}_1 \mathbf{e}_2 \mathbf{e}_3 = \mathbf{e}_2 \mathbf{e}_3 \mathbf{e}_2 \mathbf{e}_3 = - \mathbf{e}_3 \mathbf{e}_3 = -1.$

Moreover, the volume element commutes with any other element of the algebra, so that it can be identified with the complex number, whenever there is no danger of confusion. In fact, the volume element along with the real scalar forms an algebra isomorphic to the standard complex algebra. The volume element can be used to rewrite an equivalent form of the basis as

Grade	Type	Basis element/s
0	Unitary real scalar
1	Vector
2	Bivector	$\{ i \mathbf{e}_{1}, i \mathbf{e}_{2}, i \mathbf{e}_{3} \}$
3	Trivector volume element

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