Position Vectors: A Geometric Explanation

[ehrenfest]

[SOLVED] position vectors

Homework Statement

Say you have two reference frames R_1 and R_2, where the origin of R_2 is far away from the origin of R_1. Then say you have a point in space P. The position vector of P in R_1 is r_1 and the position vector of P in R_2 is r_2.
What I am confused about is that vectors are supposed to be tensors which are supposed to be geometric objects that are invariant under coordinate transformations, right? Only their components change. But when you look at r_1 it is not the same as r_2. I don't understand that?

This is from Wikipedia: "The intuition underlying the tensor concept is inherently geometrical: as an object in and of itself, a tensor is independent of any chosen frame of reference."

What am I missing?

Homework Equations

The Attempt at a Solution

 

[NateTG]

The notion of position vector is not coordinate independent. (When you talk about a 'position vector' you're talking about a vector from the origin to some point.)

If we talk about vectors in terms of the vector's origin and destination, you will find that
<d_x-o_x,d_y-o_y,d_z-o_z>
where the d's and o's are the coordinates of the vector's origin and destination, you will find that the terms are invariant over translation.

 

[ehrenfest]

Please confirm the following statement:

a position vector is a misnomer; it is not really a vector since it is not really a tensor since it is not really a geometric object that is independent of the IRF

 

[NateTG]

Please confirm the following statement:

a position vector is a misnomer; it is not really a vector since it is not really a tensor since it is not really a geometric object that is independent of the IRF

http://mathworld.wolfram.com/Vector.html

No, that's incorrect. Rather, I meant that if \vec{v} is a position vector in one reference frame let's call that reference frame A, that doesn't guarantee that \vec{v} is a position vector in some other reference frame B.

Let's say we have \vec{v}=\vec{AB}. Now, if A happens to be the origin, then we can call \vec{v} a position vector, but when we change reference frames, A might no longer be the origin, and the \vec{v}[/itex], while still a vector, is not a position vector anymore.<br /> <br /> N.B.: I can&#039;t recall seeing any formal definition of position vector, but in my experience it refers to &#039;vector with it&#039;s tail/base/start at the origin.&#039;

 

[ehrenfest]

i see. thanks.