What exactly is a position vector?

[korters]

What exactly is a position vector?

Does a position vector ALWAYS have it's initial point at the origin (0,0)? What if there was an equivalent vector from P1 (NOT the origin) to P2 that has the same magnitude and direction as a vector with its initial point at the origin? Obviously these vectors would be equal, but would that mean the 1st vector that DIDN'T have its initial point at (0,0) is a position vector as well?

Thanks

 

[HallsofIvy]

You are actually using the word "vector" in two different ways. Strictly speaking a vector is an equivalence class of pairs of points. A pair (P1,Q1) is equivalent to a pair (P2, Q2) if and only if P1<sub>x^{- Q1x}= P2x[/sup]- Q2x and P1y- Q1y= P2y- Q2y. That is the sense in which the "vector" from (0,0) to (1, 1),say, is "equal to" the "vector" from (1, 2) to (2, 3). But when we talk about the "position vector" of a point we are talking about a single pair of points in that equivalence class: The pair of points (P,Q) with P= (0,0). So a "position vector" isn't a "vector" in the more general sense: once you stop working in Euclidean, "flat", space, the whole concept of "position vector" disappears.

The position vector of a point, P, is the vector having its initial point at (0,0) and final point at P. A vector having the same direction and length as OP but going</sub>