Geometric Product

The Geometric Product is the fundamental operation of Geometric Algebra. It is a Bilinear Product. We define Multivectors as Vectors closed by this product, which make them an Algebra over a Field.

The Geometric Product is the sum of the Inner Product and the Outer Product:

$\vec{a}\vec{b}=\vec{a}\cdot \vec{b}+\vec{a}\wedge \vec{b}$

Calculation

The basis vectors ${\vec{e}}_i$ satisfy:

${\vec{e}}_i^2=1{\vec{e}}_i{\vec{e}}_j=-{\vec{e}}_j{\vec{e}}_i(i\ne j)$

To compute $\vec{a}\vec{b}$, expand in the basis and distribute. For example in ${\mathbb{G}}^2$ with $\vec{a}=a_1{\vec{e}}_1+a_2{\vec{e}}_2$ and $\vec{b}=b_1{\vec{e}}_1+b_2{\vec{e}}_2$:

\vec{a}\vec{b} &= a_1 b_1 \vec{e}_1^2 + a_1 b_2 \vec{e}_1 \vec{e}_2 + a_2 b_1 \vec{e}_2 \vec{e}_1 + a_2 b_2 \vec{e}_2^2 \\ &= (a_1 b_1 + a_2 b_2) + (a_1 b_2 - a_2 b_1) \vec{e}_1 \vec{e}_2 \\ &= \vec{a} \cdot \vec{b} + (a_1 b_2 - a_2 b_1) \vec{e}_{12} \end{align}$$ where $\vec{e}_{12} = \vec{e}_1 \vec{e}_2$ is a bivector