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Do truly absolute directions exist? If not, what directions exist independent of coordinate systems? I feel that, until I understand the answers to these questions, I will not fully understand how vectors exist independent of coordinate systems.

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  • $\begingroup$ Do you fully understand how a coordinate system can exist without vectors? $\endgroup$ – Bill N May 9 at 22:57
  • $\begingroup$ Coordinates constitute points, not directed line segments. $\endgroup$ – Vinny May 10 at 0:30
  • $\begingroup$ I still feel that the question is not well-posed, attracting many answers but not the one you are looking for. What do you mean with direction independent of coordinate systems? What do you mean with vectors, which exist independent of coordinate systems? Are you referring to a specific theory? (Classical mechanics, General relativity...) Where have you heard/read these statements? $\endgroup$ – Cream May 10 at 11:39
  • $\begingroup$ If I understand correctly, in Classical Mechanics a true vector possesses a magnitude and direction that do not depend a coordinate system. However, I do not understand how one may arrive at these values for a vector in an absolute way. $\endgroup$ – Vinny May 10 at 17:28
  • $\begingroup$ What do you mean by a "true vector"? One that exists in physical reality? Lets say we have a physical system with some rod that has a length and an orientation. Then we can describe this system in many different coordinate systems (e.g. rotated or translated with respect to each other). The vector, describing the rod, would then depend on the choice of coordinate system, but exists independent of that choice. Would that be a true vector? Or do you mean something else? $\endgroup$ – Cream May 11 at 7:34
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No. If what you mean by "directions" is a set of independent (unit) vectors whose span is the vector space, then any invertible linear transformation will give you a different set of "directions."

To address your second point which is to understand what vectors are, it's best to begin with axiomatic vector space theory (c.f. Wikipedia). Vectors are elements of a vector space. Once you understand the properties of vector spaces (and their structure-preserving maps), you can understand that "directions" (more commonly called bases) are any linearly-independent spanning set on the vector space, and is moreover not unique (the sets are related by these maps). In fact for a vector space of dimension $n$, any set of $n$ linearly-independent vectors constitutes a basis. A coordinate system is then the choice of one such basis: an arbitrary vector in this vector space can then be expressed as a linear combination of such basis elements (by the spanning property), and the resulting coefficients of such a combination are the coordinates of the vector.

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One example is the direction between two stars in a constellation. The relative position between then changes so little in a timelife that can be taken as fixed.

But for that ones like ursa major in the northern hemisphere or crux in the southern one, all the group rotates along one night. A direction between 2 stars parallel to the horizon at $9$PM becomes perpendicular six hours later.

Nothing moved with the vector, but our (informal perception) coordinate system has rotated.

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