Let $\sigma_x$, $\sigma_y$, $\sigma_z$ be the standard Pauli matrices.
Prove, if $\alpha \cdot \sigma = \alpha_x \sigma_x + \alpha_y \sigma_y + \alpha_z \sigma_z$, that $\alpha \cdot (\sigma \beta) \cdot \sigma = \alpha \cdot \beta + i \alpha \times \beta \cdot \sigma$.
What does $\times$ represent in this question, cross product or multiplication ?. And also what about $\alpha \cdot \sigma\beta$ , is it $\alpha_x \sigma_x \beta_x + \alpha_y \sigma_y \beta_y+ \alpha_z \sigma_z \beta_z$. If yes, how can we do the second dot operation $\cdot \sigma$ ?