How does rotating a coordinate system affect vector direction and components?

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Discussion Overview

The discussion revolves around the effects of rotating a coordinate system on the direction and components of a vector. Participants explore the implications of changing coordinate systems in relation to physical representations of vectors, using examples to illustrate their points.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants propose that rotating a coordinate system does not change the physical direction of a vector, as the vector remains pointing towards the same physical target, such as the Moon.
  • Others argue that while the vector's direction remains unchanged, its components will differ due to the new orientation of the coordinate system.
  • A participant introduces an analogy involving a compass needle, suggesting that the needle's direction remains constant despite the rotation of the compass casing, emphasizing the need for a coordinate system to describe vectors unambiguously.
  • Another participant challenges the necessity of a full coordinate system, suggesting that a single reference direction is sufficient for describing a vector's direction, as illustrated by an example of a rocket's velocity towards Earth.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of a complete coordinate system for vector description, indicating that multiple competing perspectives remain unresolved.

Contextual Notes

Some statements rely on assumptions about the nature of vector representation and the definitions of coordinate systems, which may not be universally agreed upon.

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If I move a coordinate system by an angle theta, why does the vector still have the same direction, but the components are different?
 
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Because you just changed the coordinate system, you did not change anything physically. If the vector pointed towards the Moon before it will still point towards the Moon if you pick another coordinate system.
 
I had another example with a compass needle pointing North regardless of how you rotate the casing of the compass on which N-S & E-W markings are drawn, but @Orodruin got there first with his example. I should add, though, that this example shows that a coordinate system is always needed before an unambiguous description of a vector can be written down.
 
kuruman said:
I should add, though, that this example shows that a coordinate system is always needed before an unambiguous description of a vector can be written down.
This is not really true. You just need a single reference direction in which the vector is pointing. For example, "the rocket is moving with a velocity of 0.5 towards the Earth".
 

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