Any vector space V over a field F by definition contains in it a rule or a operation called scalar multiplication which associates with each scalar 'c' in F, a vector 'c$.\alpha$' in V such that,
i) 1.$\alpha$ = $\alpha$
ii)($c_1*c_2).\alpha$ = $c_1.(c_2.\alpha)$
are satisfied where 1 is the multiplicative identity of the field of scalars F.
So, there must be some natural interpretation of why we define a vector space in a way that the field of scalars if considered as a group with multiplication '*' (ignoring the addition operation since here, $(c_1+c_2).\alpha \neq c_1.(c_2.\alpha)$) acts on the so called set of vectors V. Here the set V before the group action can also be considered as a commutative group with the addition operation.
Does this mean that the orbits corresponding to this group action on V is nothing but all the possible one dimensional subspaces of V? (Example: If V is the coordinate plane and the scalar field is the set of real numbers, then the orbits of V are all the lines passing through the origin?)
Lastly, then what would be the relevance of these remaining rules:
iii)$c.(\alpha+\beta) = c.\alpha+c.\beta$
iv)$(c_1+c_2).\alpha = c_1.\alpha+c_2.\alpha$