When reading introductory texts on geometric algebra, the author usually introduces two kinds of products and provides a geometric interpretation for understanding them:
- the dot product or inner product $\vec{a} \cdot \vec{b}$ represents projecting $\vec{a}$ on to $\vec{b}$ and scaling by the magnitude of $\vec{b}$, or vice versa.
- the wedge product or outer product $\vec{a} \wedge \vec{b}$ represents the bivector or “plane segment” produced by displacing $\vec{a}$ along $\vec{b}$
Both products can be visualized, and have properties that follow intuitively from their geometric interpretations like commutativity or anti-commutativity and relationships with $\sin$ and $\cos$.
However, when the geometric product $\vec{a}\vec{b} = \vec{a} \cdot \vec{b} + \vec{a} \wedge \vec{b}$ is introduced, there is no explanation of what it even means to add a scalar and a bivector or how you would visualize that compound object, and therefore, it isn't as easy to reason intuitively about the properties of the geometric product.
Is there a good way of visualizing such a multivector? And what is the purpose of combining the two products like this when they seem to be represent separate ideas?
Edit In response to a comment, I'm also curious about Clifford algebra: what is it, what is its relationship to geometric algebra, and what insights it does it provide?
A product $\mathbf{uv}$ of vectors can be written $r(\cos\theta + \mathbf{i}\sin\theta) = r\exp{(\mathbf{i}\theta)}$, where $\mathbf{i}$ is the unit pseudoscalar of the plane containing $\mathbf{u}$ and $\mathbf{v}$ and $r = |\textbf{u}||\textbf{v}|$ and $\theta$ is the angle between the vectors.
This is interpreted geometrically as an arc of a circle of radius $r$ subtending the angle $\theta$. The arc can be slid around the circumference without changing the interpretation, just as a vector can be moved parallel to itself without changing it.