The geometric product is the fundamental operation of geometric algebra (GA). It unifies the dot product (scalar) and the wedge product (bivector) into a single, associative, non-commutative operation. All other GA concepts (inverses, rotations, duals, etc.) are derived from this single definition.
Let \(\mathcal{V}_n \subset \mathbb{R}^n\text{.}\) For any two vectors \(\mathbf{u}, \mathbf{v} \in \mathcal{V}_n\text{,}\) the geometric product is defined as
\(\mathbf{u}\cdot\mathbf{v}=|\mathbf{u}| |\mathbf{v}| \cos{\theta}\) is the symmetric (scalar) component, and provides information about projection and magnitude.
\(\mathbf{u}\wedge\mathbf{v}= |\mathbf{u}| |\mathbf{v}| \sin{\theta} \, \mathbf{i}\) denotes the antisymmetric (bivector) component, and corresponds with an oriented area that specified by the area of the plane segment spanned by \(\mathbf{u}\) and \(\mathbf{v}\text{.}\)
Here, \(\mathbf{i}\) is a bivector representing the unit pseudoscalar of a plane. Importantly, it satisfies \(\mathbf{i}^2=-1\text{,}\) providing a natural geometrical interpretation of complex numbers.
