Exponential of a Pauli vector ($e^{-ia\mathbf{n}\cdot \mathbf{\sigma }}=I\cos a-i\mathbf{n}\cdot \mathbf{\sigma }\sin a$)

For a real unit vector $\mathbf{n}$, the Pauli matrix multiplication law gives $(\mathbf{n}\cdot \mathbf{\sigma }{)}^2=I$. Separating the even and odd powers in the matrix exponential proves the displayed identity. For spin one-half, the physical rotation through angle $\theta$ has $a=\theta /2$.