Of course one can speak of the "component" of a vector without reference
to a coordinate system.
Given a vector A and a direction b (usually defined by a unit vector b
in that direction), the component of A along b is given by the scalar
product A dot b. Scalar products are independent of the coordinate
system.
We *usually* choose our direction b to be one of our basis vectors which
we *usually* choose to be a unit vector which we *usually* choose to lie
along an axis of our coordinate system. But we don't have to. We could
choose basis vectors which are not unit vectors. We could choose basis
vectors which are neither unit vectors nor which lie along coordinate
axes. Heck, we could even talk about components of vectors along
directions which are not basis vectors, nor unit vectors, nor lie along
a coodinate axis.
Glenn A. Carlson
St. Charles Community College
gcarlson@stchas.edu