As far as I understand it, the notion of direction is not an intrinsic property of a vector in an abstract vector space; if the space is equipped with an inner product then one can determine angles between vectors and hence then gain a notion of a vector have direction (as well as a magnitude, as an inner product implies that the vector norm exists). Have I understood this correctly?
If so, when defining tangent vectors on a manifold, is the motivation for considering them as equivalence classes of curves passing through a given point (with equal derivatives, to first-order, in some coordinate chart), that the directional derivatives that arise from this approach satisfy the vector space axioms and thus are sufficient to unambiguously define vectors at each point on the manifold?
My confusion stems from the fact that if one chooses a curve passing through a particular point on a manifold and then takes the directional derivative on some differentiable function $f$ defined on $M$, then aren't we implying through this that defined via such curves intrinsically have a notion of direction at each point on the manifold?