Question about Leo Dorst book on Geometric algebra for Computer Science

Question

In section 7.4.3 Dorst brings a result that in Euclidean and Minkowski spaces, a bivector $B$ can be written as a sum of commuting 2-blades, and therefore $e^B=e^{\mathbf B_1}\cdots e^{\mathbf B_k}$. Each of $e^{\mathbf B_i}$ is a rotor, so it is a geoemtric product of two unit vectors, say $\mathbf b_{2i-1}\mathbf b_{2i}$. I understood that $e^{\mathbf B_i}=\mathbf b_{2i-1}\mathbf b_{2i}$. But Dorst claims further that $\mathbf B_i=\mathbf b_{2i-1}\land \mathbf b_{2i}$. This I don't understand. Can somebody help here?

Answer 1

The relation $\mathbf{B}_i = \mathbf{b}_{2i - 1}\wedge\mathbf{b}_{2i}$ is not, in fact, a claim. In that section, it is actually how those vectors are defined. Each of the $\mathbf{B}_i$ entities in the expression given in $(7.16)$ are blades, which means the decomposition as the wedge of two vectors is possible. The question you may be asking is why $e^{\mathbf{b}_{2i - 1}\wedge \mathbf{b}_{2i}} = \mathbf{b}_{2i - 1} \mathbf{b}_{2i}$. This follows directly from the Taylor expansion and one additional constraint given, which is that the $\mathbf{b}_i$ vectors have unit length.