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?
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Cylindrical and spherical unit vectors are independent too. Their directions are dependent upon not just the coördinate system, but the particular coördinate location too. – naturallyInconsistent Jul 21 '23 at 02:46
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Duplicate : Unit vectors in the cylindrical coordinate system as functions of position. – VoulKons Jul 21 '23 at 05:53
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Does this answer your question? Unit vectors in the cylindrical coordinate system as functions of position – VoulKons Jul 21 '23 at 05:53
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In non Cartesian coordinate systems we can still talk about a basis of orthogonal unit vectors. These orthonormal frames are typically "moving". The matters are explicitly discussed here. How the cross product behaves in cylindrical and spherical coordinates I find particularly interesting. See here – Kurt G. Jul 21 '23 at 09:12
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and here. – Kurt G. Jul 21 '23 at 09:12
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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.
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