The set
that contains the zero vector, so
\[\begin{equation*}
\vec{0} \in V,
\end{equation*}
\]
in conjunction with the vector addition operation
\[\begin{equation*}
\forall \vec{u}, \vec{v} \in V \> : \> \vec{u} + \vec{v} \in V
\end{equation*}
\]
and scalar multiplication operation
\[\begin{equation*}
\forall a \in F, \> \forall \vec{v} \in V \> : \> a \vec{v} \in V
\end{equation*}
\]
which is the one and only
link between vector addition and scalar multiplication
in the vector space.