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

[Mathematicsresear]

If I move a coordinate system by an angle theta, why does the vector still have the same direction, but the components are different?

 

[Orodruin]

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.

 

[kuruman]

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.

 

[Orodruin]

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".