Is position vector really a vector if it depends on the origin?

[amit_thakur]

vectors were introduced because they help physicists specify laws without reference to any
particular coordinate system...can we then say that position vector is really a vector because
after all it depends on where we choose our origin?

 

[JesseC]

Vectors are necessary to describe things which have both direction and magnitude such as velocity, acceleration, torque, momentum.

The position vector does change magnitude and direction depending on where you place your origin, you're right. This doesn't mean it's no longer a vector... a vector is just a mathematical object. You still describe the position as components multiplied by unit vectors no matter where the origin.

 

[K^2]

vectors were introduced because they help physicists specify laws without reference to any
particular coordinate system...can we then say that position vector is really a vector because
after all it depends on where we choose our origin?

Vector itself does not depend on the coordinate system. What you write down as <x, y, z> is just a representation of a vector in a specific coordinate system. The representation changes with coordinate system, but the vector does not.

Vector itself is an abstract object that simply follows a set of axioms outlined here.

So if you have points at vector coordinates a and b, the relative position of b with respect to a is always b-a. Again, the representation can change from coordinate system to coordinate system, but the relative position will never be anything other than b-a.

In other words, what is preserved under coordinate system transformation is the relationships between the vectors, rather than their representations, which is significantly more important when you are doing physics independent from coordinate system.