Tangent vectors as directional derivatives

["Don't panic!"]

I've become confused in studying differential geometry as to whether direction is an intrinsic property of vectors or not?! My understanding from studying abstract vector spaces is that it isn't (there is no reference to a notion of direction or magnitude of a vector in the vector space axioms) and that one can only gain a notion of magnitude and direction when one introduces a norm and inner product for a given space?!

 

[Fredrik]

I've become confused in studying differential geometry as to whether direction is an intrinsic property of vectors or not?! My understanding from studying abstract vector spaces is that it isn't (there is no reference to a notion of direction or magnitude of a vector in the vector space axioms) and that one can only gain a notion of magnitude and direction when one introduces a norm and inner product for a given space?!

Depends on what you mean by "direction". For any non-zero vector $x$ I think it's very natural to think of a curve of the form $t\mapsto tx$ as singling out a direction in that vector space. And since there's one such curve for each non-zero vector, we might as well say that each non-zero vector identifies a direction.