$\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}$

$\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u}\cdot\mathbf{v} + \mathbf{u}\cdot\mathbf{w}$

$(c\mathbf{u}) \cdot \mathbf{v} = c(\mathbf{u}\cdot\mathbf{v})$

$\mathbf{u} \cdot \mathbf{u} \geq 0$