Let $M$ be a manifold of dimension $m$ and $p\in M$, then $T_p(M)$, the tangent space of $M$ at the point $p$, is also of dimension $m$.
A vector field $X$ on $M$ is a map that associates to each point $p\in M$ a vector $X(p)$ of $T_p(M)$, so $X(p)$ has $m$ coordinates.
A line field $P$ on $M$ is a map that associates to each point $p\in M$ a vector subspace of dimension one $P(p)$ of $T_p(M)$, so $\dim(P(p))=1$.
It is said, as an example, if $X$ is a non-singular vector field on $M$, we can define a line field $P$ on by letting $$P(p)= \mathbb{R} . X(p)$$
How could this be done?
All multiples of vector $X(p)$ gives you that subspace of $T_p(M)$.