I started looking into Clifford Algebras, but I think I am conceptually missing some points. Here is how I understand the geometric product now (leaving out coordinate system independence and the generalized definitions of the products for brevity):
Let's say we have a Clifford Algebra $Cl(\mathbb{R^n})$ with $e_1...e_n$ as orthonormal basis, where $e_i^2 = 1$ and $e_ie_j = 0$ for $i \neq j$ for simplicity.
The geometric product is associative and distributive over addition. Also for 1-vectors the geometric product can be represented by $ab = a \cdot b + a \land b$. Where $a \cdot b = b \cdot a$ and $a \land b = -b \land a$, which gives $a \cdot b = 0$ if $a \perp b$ and $a \land b = 0$ if $a \parallel b$.
Now, I have the following problem, which I think gives the most concise way of representing my question:
Given $a = e_1$ and $b = e_2 \land e_1$, what is $ab$?
With what I have above, I can solve this in the following four ways, giving me different answers:
$$ab = e_1e_2 \land e_1 = 0 \land e_2 = 0$$ $$ab = -e_1e_1 \land e_2 = -1 \land e_2 = -e_2$$ $$ab = e_1e_2 \land e_1 = (e_1 \cdot e_2 + e_1 \land e_2) \land e_1 = e_1 \land e_2 \land e_1 = 0$$ $$ab = -e_1e_1 \land e_2 = -(e_1 \cdot e_1 + e_1 \land e_1) \land e_2 = -1 \land e_2 = -e_2$$
What point am I missing here?
