Are vectors independent of reference frames?

In summary, the conversation discusses the notation used for vectors and coordinates in different reference frames. It is mentioned that vectors are independent of frames, but their components are not, and tensors are used to account for changes in components when frames are rotated or deformed. In a mathematical context, the term "vector" may not always refer to a tensor.
  • #1
TonyEsposito
37
5
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
 
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  • #2
TonyEsposito said:
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.
 
  • #3
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.)
 

Related to Are vectors independent of reference frames?

1. What are vectors?

Vectors are mathematical objects that have both magnitude (size) and direction.

2. How do vectors relate to reference frames?

Vectors are independent of reference frames, meaning they have the same magnitude and direction regardless of the chosen coordinate system.

3. Can vectors change in different reference frames?

Yes, the components of a vector may change in different reference frames, but the vector itself remains the same.

4. What is the difference between a vector and a scalar?

A scalar is a quantity that only has magnitude, while a vector has both magnitude and direction.

5. Why is it important to understand the independence of vectors from reference frames?

Understanding the independence of vectors from reference frames is important in various fields, such as physics and engineering, where vector quantities are used to describe physical phenomena. It allows for consistent and accurate analysis and calculations, regardless of the chosen coordinate system.

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