Unit Vectors and Coordinate Systems

Question

Are $x$, $y,$ and $z$ unit vectors coordinate system independent like any true vector? The same question can be asked of cylindrical and spherical unit vectors. I know that one may perform the same vector operations on these vectors as with any other vector. But, do the directions of these unit vectors not dependent on particular coordinate systems?

Answer 1

The $\hat x,\, \hat y,\ \hat z$ unit vectors define a coordinate system.

But the answer is yes. Like with any vector, the components of the vector may change with a new coordinate system, but the magnitude and direction will not.