I’m going through the book Geometric Algebra for Physicists, by Doran and Lasenby, and I have found myself lost when the authors use the grade operator to change between products and switch the order. For example
A \cdot(\boldsymbol{x} \wedge(\boldsymbol{x} \cdot B))=\langle A \boldsymbol{x}(\boldsymbol{x} \cdot B)\rangle=\langle(A \cdot \boldsymbol{x}) \boldsymbol{x} B\rangle=B \cdot(\boldsymbol{x}\wedge(\boldsymbol{x} \cdot A))
in page 75, or
\begin{aligned} \left\langle A_{r} \mathrm{~F}\left(B_{r}\right)\right\rangle &=\left\langle A_{r} B_{r}\right\rangle+\alpha\left\langle A_{r}\left(B_{r} \cdot f_{1}\right) \wedge f_{2}\right\rangle \\ &=\left\langle A_{r} B_{r}\right\rangle+\alpha\left\langle f_{2} \cdot A_{r} B_{r} f_{1}\right\rangle \end{aligned}
on page 110.
Can someone explain the rationale behind these examples and/or provide a source where these operations are explained?
Thanks.