Consider a single particle moving around the origin in a circle. If the particle's coordinates are $(x,y)$ at some time, it will reach $(x,-y)$ after some time as they lie on the circle.
The position vector of the particle is rotating about the origin. Now rotation of a vector by an angle $Z$ clockwise is equivalent is to rotating the coordinate axes anticlockwise by $Z$.
The components in the new axes rotated anticlockwise are given by : $$ x' = x\cos(Z) + y\sin(Z)$$ $$ y' = -x\sin(Z) + y\cos(Z)$$
These are also the component transformations if I rotate the vector clockwise and keep my axes the same.
The question is : By rotation of the axes , we cannot transform the vector $(x,y)$ into $(x,-y)$ since there is no angle $Z$ that satisfies the equations for this pair ( x and y not 0 ). But on rotating the vector we get $(x,-y)$. This seems to be a paradox. What am I missing here?