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I rather like the use of the vectors A_x and A_y, as they provide a nice
geometric approach to introducing the idea of components. One has probably
just finished defining vector addition geometrically (head-to-tail and
parallel transport). Its then natural to define component vectors as those
unique vectors parallel to the coordinate axes that add up to the vector A.
One can then write:
A_x vector = (A_x) i_hat etc