Components of vectors (polar coordinates)

[PFuser1232]

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?

 

[jedishrfu]

You can transport a vector around and it should be independent of its position in N-dimensional space. You would describe it as having a length and a direction and transporting it about would not change that.

There was a thread here discussing polar coordinate unit vectors which may help:

https://www.physicsforums.com/threads/polar-unit-vectors-form-a-basis.703992/

 

[Chestermiller]

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)

The vector itself, as expressed in terms of the sum its components times the coordinate system unit vectors will not change, but, if the coordinate system is curvilinear (so that the directions of its unit vectors vary with spatial position), then the components expressed with respect to these unit vectors will change. Cartesian coordinates are not curvilinear, so its unit vectors are pointing in the same directions at each spatial position, and the components of a vector expressed with respect to this coordinate system do not change.

Chet