# A vector should more be thought an identity of an entity in space rathar than magnitude + direction?

Can I say that vector is more like a "unique identity" of an entity in space rather than calling it an entity with magnitude and direction ?

For example a line. A vector $(10,10,0)$ is the identity of a unique line that starts from $(0,0,0)$ and ends up at $(10,10,0)$ .

Can I apply this notion everytime, everywhere in math and physics ?

In fact, in maths first you define vector space as a module over a field (take a look at wiki article). And only then you define a vector as an element of a vector space. Moreover, in most cases the "direction" is not a notion one can easily describe. Take, for example, Lebesgue spaces, or any banach space of infinite dimension.

In physics, mostly in analtical mechanics, it's not uncommon to see a reasoning of the type "let's take a vector from point $A$ to point $B$", which begs to define vectors as a class of equivalence (magnitude and direction). All such approaches are equivalent to a formal one, so it's up to you to chose one that makes the reasoning concise and clear.

The description of vectors as having magnitude and direction is misleading in that it applies only to some vector spaces and not to others. I think, though, that "unique identity of an entity" is even worse, since it strikes me as an essentially meaningless phrase.

• what vector spaces don't use direction to specify itself ? Commented Jul 12, 2013 at 18:10
• @VishwasGagrani For example, the vector space of all continuous real-valued functions on $[0,1]$ doesn't involve magnitude and direction in the usual sense of those words. Commented Jul 12, 2013 at 18:40

In 2D and 3D, I think it's quite reasonable to define a vector as a "directed arrow", or as a displacement of position, or as "a thing that has magnitude and direction (but no fixed position)". You can then show (roughly) that these things form a vector space by checking the mathematical axioms via physical reasoning.

In more abstract settings, a vector is typically defined as an element of a vector space, as the other answer said. This always seems a bit circular, to me. So, if you only care about 2D and 3D, I'd suggest using the less formal (and more physical) notions that I gave above.

Your idea of identifying a vector with a line segment is not quite correct. The line segment has a fixed position in 3D space, whereas vectors have no position.

Well, i got my answer through another source. So, vectors are not any unique identity on a graph. They are have a magnitude and direction. For example a vector of 5 magnitude and subtending an angle of 30 degree. There are infinite possibilities of such vectors on a graph. Thus vector is not an identity of a unique entity.

• One of possible approaches to vectors is to introduce classes of equivalence. Let's for simplicity take $\mathbb R^2$. On the set of all segments $\vec {AB}$ joining two points on our plane, we say $\vec{CD}\equiv \vec{FG}$ if they have the same direction (angle) and magnitude. We, of course, need to define "direction", prove that it's indeed equivalence and show that these classes of equivalence form a vector space, which we conventionally call $\mathbb R^2$. Commented Jul 12, 2013 at 13:37