If one considers a smooth manifold, then it is possible to define (tangent) vectors in the tangent space to each point on that manifold without having to introduce any additional structure such as a metric.
As such (tangent) vectors can be defined in terms of equivalence classes of curves (with a tangent vector at a given point defined as an equivalence class of curves through that point that are all tangent to one another at that point), is it correct to say the notion of a vector having "direction" is meaningful without needing to introduce a metric, or do the two come hand in hand?
I ask as I know that vectors (in general) do not have "length" without first defining a metric, however, I'm not sure that the same can be said for direction? I think I'm slightly confused with the notion of defining vectors on a manifold and more abstract vector spaces, for example, the set of polynomials form a vector space however it doesn't seem to make sense (at least to me) that these polynomials have "direction"?!
Any help would be much appreciated.