I would ask you to comment on the two ideas about how vectors can be represented on the plane.

1) Vectors are used to be elements of a vector space. So, if my vector space is named $\mathbb{R}^2$, then my vectors have no choice and have to be points $(x,y)$ on the plane. So, let me take some vector $a \in \mathbb{R}^2$ and some positive real number $r$. Then, $B(a,r)=\{z \in \mathbb{R}^2 \mid \Vert a-z \Vert < r\}$ will be my open ball with a center $a$ and and radius $r$. It consists of all the points, which distance to $a$ is smaller than $r$.

2) Vectors are used to be drawn as arrows in the plane. So, if I want to depict a vector with the coordinates $(x,y)$, I put my pencil to the $0$ and draw an arrow, which points directly to the point I have - namely, to $(x,y)$. But, let me name this vector as $a$. Then, $B(a,r)=\{z \in \mathbb{R}^2 \mid \Vert a-z \Vert < r\}$ will be my open ball with a center (?) $a$ and radius $r$. It contains of all those vectors which length is smaller than $r$. As I used to draw all the vectors from the zero, - so, I would better draw a circle with the radius $r$ around zero. Then, all the vectors pointing somewhere inside this circle will be elements of my flat but nice ball.

So, even that I hope that I represent both ideas correctly, I still do not understand the difference in these representations. Except the obvious one: point is not equal arrow. Clear.

In my course of analysis we used to set vectors as the elements of the given vector space. So, I am pretty clear about the 1. representation. On the other hand, the triangle unequality becomes perfectly clear if you represent vectors of $\mathbb{R}^2$ as arrows on the plane. But then, was is meant by "arrow to be a center"? Imagine I draw an arrow with the length $\le r$ that starts not from $0$. Will it be still in $B(a,r)$?

So, I would ask you to share your experience or ideas about how these representations should be used correctly, and why do we have two (or more?..) of them. What is the idea behind each of them? Can you correct something in my text? Have I made any mistake describing these two ways?

(except my English grammar - I am sorry for my French...)

  • 1
    $\begingroup$ Your interpretation in (2) is mistaken. $B(a, r)$ is all the arrows whose heads are within $r$ of the head of $a$. So it's still a circle around $a$, and a whole lot of arrows pointing into that circle. At this point it might be convenient to stop drawing vectors as arrows and just draw points instead. $\endgroup$
    – Arthur
    Sep 27 '17 at 12:58

It's something that has already been discussed here.

As you said, a 2D point can be seen as an element of a $\mathbb{R}^2$ 1); but it can as well be seen as the translation that transforms the origin of the vector space to the 2D point your interpretation 2). The 2 have different meaning but bring the same information.

I personally find that for some things the transformation interpretation is better(especially when the norm is involved, like for the triangle inequality); but in other cases (like for topological concepts), I prefer to think of it only as a 2D point (your first interpretation).

  • $\begingroup$ Do you mean something like "creating an affine subspace"? Or how should one interprete the translations you describe? $\endgroup$
    – Kirill
    Sep 27 '17 at 18:20
  • $\begingroup$ No it's not what meant. What I meant is that since the group of translations in $\mathbb{R}^2$ is isomorphic to $\mathbb{R}^2$ . You can choose the angle you want to see an object or its matching translation. So it's up to what you suit you best :) $\endgroup$
    – Leo
    Sep 27 '17 at 20:03

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