Per its algebraic properties,

• for all vectors \(\{\vec{u}, \vec{v}\} \subseteq V\) and
• for all scalars \(\{a, b\} \subseteq F\),

it:

• commutes, so

  \(a \vec{v} = \vec{v} a\);

• associates, so

  \((a b) \vec{v} = a (b \vec{v})\);

• distributes over vector addition, so

  \(a(\vec{u} + \vec{v}) = a \vec{u} + a \vec{v}\); and

• has exactly one multiplicative identity: \(1\), so

  \(1 \vec{v} = \vec{v}\).