For all vectors \(\{\vec{u}, \vec{v}, \vec{w}\} \subseteq V\), it

• commutes, so

  \(\vec{u} + \vec{v} = \vec{v} + \vec{u}\),

• associates, so

  \((\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})\),

• has exactly one additive identity: the zero vector \(\vec{0}\), so

  \(\vec{v} + \vec{0} = \vec{v}\),

• has exactly one additive inverse: \(-\vec{v}\), so

  \(\vec{v} + (-\vec{v}) = \vec{0}\).