I have always been under the impression that a vector is not "fixed" in space. Given any vector, we could just move it around and it would still have the same components (in a cartesian coordinate system). What confuses me, however, is how we define the components of a vector in polar coordinates. If we "move the vector around", we don't seem to get the same components (in polar coordinates). Does that mean that when talking about vectors in a polar coordinate system, we are not allowed to "move" the vector around?