[cat9]

"But for physics, it is absolute essential that you are also able to think about vectors in terms of coordinates if you are to connect them to the real physical world, which has rulers and T-squares and so on. Physicists are trying to understand the world, not the abstraction."

This breaks down the second you encounter, for example: Fourier transforms, wave equations, or hyperbolic geometry. Physicists are trying to understand the world, which in many cases does not comport itself in a way which is considerate of our naive intuition about mechanics and geometry.

The nth dimensional component of a vector still has a specific meaning within the context of the parent model, but it's far easier if you stop trying to force that and learn to deal with vectors as vectors instead. This allows for abstractions which are portable between the very many places where vectors are useful, instead of trying to force each value to fit an ad hoc cognitive model, with whatever flaws and conceptual limits that it introduces.

[jessriedel]

Like jblow, you are under the mistaken impression that I am downplaying the importance of the geometric interpretation.