The coordinates of v must be transformed into the new coordinate system, but the vector v itself, as a mathematical object, remains independent of the basis chosen, appearing to point in the same direction and with the same magnitude, invariant to the change of coordinates. The contravariant transformation ensures this, by compensating for the rotation between the different bases. If we view v from the context of the radial coordinate system, it appears to be rotated more clockwise from the basis vectors er and eφ. compared to how it appeared relative to the rectangular basis vectors ex and ey. Thus, the needed contravariant transformation to v in this example is a clockwise rotation.