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Subsubsection Clifford Algebras
Subsubsection \(\operatorname{Cl}(2)\)
Clifford algebra of 2D Euclidean space.
Basis Elements.
\(\{ 1 \}\)
Corresponds with a scalar magnitudes.
\(\{ \mathbf{e}_1, \mathbf{e}_2 \}\)
Corresponds with a vector (i.e., directional information).
\(\{ \mathbf{e}_{12} \}\)
Usually called
bivectors or
wedge products . The unit bivector is the pseudoscalar element in
\(\mathrm{Cl}(2)\text{.}\)
Subsubsection \(\operatorname{Cl}(0,3)\)
Clifford algebra of 3D Euclidean space.
Basis Elements.
\(\{ 1 \}\)
Corresponds with a scalar magnitudes.
\(\{ \mathbf{e}_1, \mathbf{e}_2 , \mathbf{e}_3\}\)
Corresponds with a vector (i.e., directional information).
\(\{ \mathbf{e}_{12}, \mathbf{e}_{23}, \mathbf{e}_{31} \}\)
Usually called
bivectors or
wedge products .
\(\{ \mathbf{e}_{123} \}\)
Usually called
trivectors .
The unit trivector is the pseudoscalar element in
\(\operatorname{Cl}(0,3)\text{.}\)
Hodge Dual.
Identity A.5.4 The Cross Product as a Dual Wedge Product.
In Clifford algebra \(\operatorname{Cl}(0,3)\) (Euclidean 3-D space), the standard vector cross product \(\vec{a}\times\vec{b}\) is equivalent to the dual of the geometric wedge product \(\vec{a}\wedge\vec{b}\text{:}\)
\begin{equation*}
\vec{a}\times\vec{b} = (\vec{a}\wedge\vec{b})\mathcal{I}^{-1}
\end{equation*}
where \(\mathcal{I}=\mathbf{e}_1\mathbf{e}_2\mathbf{e}_3\) is the unit pseudoscalar in \(\mathcal{Cl}(0,3)\text{.}\)
