Are vectors independent of reference frames?

[TonyEsposito]

Ok, this is the notation I am going to use in this thread: uppercase means vectors, while "[V]c" means coordinates of vector V in frame c.
I'm reading from a book: i have a reference frame "a" and a reference frame "b" rotated with respect to "a", the vector connecting the frames origin is R.
We are tracking a particle having radius vector X in frame a and X' in frame b, the book says that the relation connecting the two vector is:

X=R+Q.V'
where Q is the rotation tensor of the two frames...but arent vectors indipendent of frames? should not the relation be simply:

X=R+V'

and only when expressed in a coordinates Q comes into play?

[X]a=[R]a+Q.[V']b
[X]b=[R]b+Q^-1.[V']a

 

[member659127]

but arent vectors indipendent of frames?

They are... But their components are not. If you rotate or stretch or shrink or deform your coordinate frame the components change which what those tensors are used for.

 

[FactChecker]

I agree, as long as the term "vector" is interpreted in the physics/tensor context. In a pure mathematical context, it is possible to call something a "vector" where it does not transform as a tensor would. (A mathematician might define something like (1,0) as a unit "vector" even though it does not transform at all.)