What is the definition of a 'true vector'?

JonDrew

In my class notes my professor defined a true vector as a vector which does not depend on origin placement. Once he defined it he went on with an example of how a vectors magnitude is conserved in two different coordinate systems.

So my question is what is the definition of a true vector? Is it just a vector whose magnitude does not depend on origin placement?

he also defined "other vectors" as vectors who's magnitude does not change upon rotation and for some reason I think he referred to these as scalers, which doesn't make sense to me.

He told us that both scalars and vectors are both tensors but I still don't see the point he was trying to make.

micromass

Either you have completely misunderstood your professor or your professor has no idea what he's talking about.

I have never heard of the terms "true vectors" or "other vectors" before and I find your post very weird (not saying that that's your fault necessarily).

I would suggest you get a basic linear algebra book such as "An introduction to Linear Algebra" by Lang and work through that.

JonDrew

I was worried about that, it is possible that I am just confused on an over simplifications then.

Thanks micro.

D H

The standard terminology is bound vectors vs free vectors.

Here's the difference in a nutshell. Imagine four distinct points in space, A, B, A', and B', such that the line segments AB and A'B' are parallel and of equal length. At least for displacement vectors, it's quite common to use a directed line segment as a representation of a vector. The key distinction: Does the start point / end point matter? With free vectors (or true vectors, or just vectors), the start and end points don't matter. [itex]\vec{AB}[/itex] and [itex]\vec{A'B'}[/itex] represent the same vector. With bound vectors, those start and end points do matter. In this context, [itex]\vec{AB}[/itex] and [itex]\vec{A'B'}[/itex] are distinct bound vectors. Given another pair of points C and C' such that BC and B'C' are parallel and of equal length, [itex]\vec{A'B'} + \vec{BC}[/itex] doesn't make sense in the context of bound vectors. It makes perfect sense in the context of free vectors (or just vectors).


Please don't shoot the messenger. This is a somewhat widespread nomenclature. I personally an not a fan of this nomenclature. Adding the qualifier "bound" implies that a "bound vectors" are a kind of vector. They're not. They fall in the class of things that are not-quite-vectors rather than being a kind of vector.

Stephen Tashi

There are such things as "pseudovectors". Perhaps that is part of what the professor was saying.

atyy

The term "free vector" refers to the mathematical structure called an affine space.
http://www.cis.upenn.edu/~cis610/geombchap2.pdf
http://cmp.felk.cvut.cz/cmp/courses/...nd-Vectors.pdf

The term "bound vector" refers to the mathematical structure called a vector space (the subject of linear algebra).

In the following, I will use "vector" in the linear algebra sense. Newtonian physics has vectors such as the velocity vector. This is a member of the tangent vector space at each spatial position. Spatial positions in Newtonian physics are not vectors - they are points in an affine space, which has no preferred origin. However, if we assign for convenience and convention a preferred spatial position as the origin of a coordinate system, then spatial positions can be described by position vectors.

fortissimo

Yes, and also covectors. Such terminology is not considered in elementary courses.