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^{th}Edition, by Arfken and Weber, Chapter 1 Vector Analysis, pages 8-9, the authors make the following statement:

"If A

_{x}and A

_{y}transform in the same way as

*x*and

*y*, the components of the general two-dimensional coordinate vector

**r**, they are the components of a vector A. If A

_{x}and A

_{y}do not show this

**form invariance**(also called

**covariance**) when the coordinates are rotated, they do not form a vector."

I understand how to use equations (1.9) and their derivations, but could anyone please explain the above statement?

Thank you so much for your help...