Affine line in a vector space ($a+kv=\{a+tv:t\in k\},v\ne 0$)

In a vector space over a field $k$, an affine line is a translate of a one-dimensional vector subspace. Its direction is that vector subspace, so multiplying $v$ by a nonzero scalar preserves the direction. Over a finite field of size $q$ it contains exactly $q$ distinct points. This usage concerns affine subsets of vector spaces, rather than the affine line as an affine scheme.