Is the position vector a real vector?

Is the position vector a real vector?
I have a hard time with this question because vectors are unchanged if I were to change my reference frame.
Example: If I place a pencil such that it points towards the door. It doesn’t matter what I define my origin to be. The pencil’s length and direction remain unchanged.
However this is not true for the position vector. Example: If I move my origin back 5 meters then the position vector changes its magnitude and direction.

Please let me know if I have been trolled here.

Indeed, position is better characterized as an affine space than a vector space.

A vector space has some operations that don’t make sense for the space of positions. Specifically, the addition of two positions doesn’t make sense (which position is the position of New York plus the position of Paris?), nor does the multiplication of a position by a real number (which position is 5 times the position of New York?).

However, the difference between two positions is a vector, and differences in position form a vector space with sensible addition and multiplication operations.

In curved spacetimes positions lose even their affine structure. In curved spacetime the best mathematical structure is a manifold. The difference between two positions is no longer a vector except locally.

Dale said:

Indeed, position is better characterized as an affine space than a vector space.

A vector space has some operations that don’t make sense for the space of positions. Specifically, the addition of two positions doesn’t make sense (which position is the position of New York plus the position of Paris?), nor does the multiplication of a position by a real number (which position is 5 times the position of New York?).

However, the difference between two positions is a vector, and differences in position form a vector space with sensible addition and multiplication operations.

In curved spacetimes positions lose even their affine structure. In curved spacetime the best mathematical structure is a manifold. The difference between two positions is no longer a vector except locally.

Thanks! I’ll share this with my friends.

WildBohr137 said:

Thanks! I’ll share this with my friends.

"It's not for me, Doctor; it's for a friend.

But you were the one to go to the trouble of asking the question. 10/10
And you will knock 'em dead with @Dale 's great answer.

sophiecentaur said:

"It's not for me, Doctor; it's for a friend.

But you were the one to go to the trouble of asking the question. 10/10
And you will knock 'em dead with @Dale 's great answer.

Haha thanks!

What is a real vector?

malawi_glenn said:

What is a real vector?

It's not one of those knock-off fake vectors that you get from Southeast Asia

malawi_glenn said:

What is a real vector?

One that the is fully automatic:

haushofer said:

One that the is fully automatic:

Yoiks, that dog running downrange almost wasn't a real dog anymore...