I am puzzled with the following problem:
given a tangent vector (a d/dx) in the Euclidean plane R^2 with "a" a dimensionless scalar, the dimensionality of this vector is, I suppose, 1/[lenght] and its norm is |a|, a dimensionless scalar. Now when I take the vector (x d/dx), it is, I suppose, dimensionless ([lenght]/[lenght]), and its norm is |x|, with dimension of [lenght].
How is it possible to get a norm of a vector dimensionless and not of dimension of [lenght]?
How to unify the treatment of these vectors?
(I could also imagine a vector 1/x d/dx, wich would be of dimension 1/[lenght]^2 and norm 1/[lenght]|)
(The peculiar situation is when we take the vector at point (a,0), lets say, then for both above vector fields, we get the vector (a d/dx) and (a d/dx), but with different dimensionality of both vector and norm.)
Thank you for any help
and wish you a nice afternoon
marfi