Angular momentum, spin etc — UPSC Physics Optional IFoS 2023 Paper II

The components of arbitrary vectors \vec{A} and \vec{B} commute with those of \vec{\sigma}. Show that (\vec{\sigma} \cdot \vec{A})(\vec{\sigma} \cdot \vec{B}) = \vec{A} \cdot \vec{B} + i\vec{\sigma} \cdot (\vec{A} \times \vec{B}).