Are Vectors Between Different Points (Same Magnitude, Same Direction) Distinct? For my entire mathematics education, points and vectors were never distinguished in my head. I've recently come across an applied math textbook (in graphics) that makes a strong distinction between these two concepts, and now my world is coming apart.
So if are working in, say, $\mathbb{R}^2$, let $P = (1,0)$ and $Q = (1,1)$. Furthermore let $O = (0,0)$ and $R = (0,1)$.
Then do we have that $\vec{OP}$ = $\vec{RQ}$? After all, it seems to me that two vectors have (i) the same magnitude (i.e., $1$) as well as (ii) the same direction (they both point straight right and are parallel with the $x$-axis).
So given this, are these two vectors -- $\vec{OP}$ and $\vec{RQ}$ -- the same?
 A: There is indeed a difference between a point in $\Bbb{R}^n$ and a vector representing a change of position in $\Bbb{R}^n$.  This notion became a major (and a bit contentious) issue in the development of the CLHEP (C++ Class Library for High Energy Physics) Vector package.
In the end, it was accepted that Point3D and Vector3D were two distinct concepts (represented by two distinct C++ classes) and:


*

*You can add a Vector3D to a Point3D (getting a Point3D) or to another Vector3D (getting a third Vector3D) but you can't add a Point3D to a Point3D.

*To retain the commutativity of the addition operator, it was deemed meaningful to do $v + p$, interpreted as assigning a starting point ("root") to a vector, thus obtaining a new Point3D which is identical to that obtained by $p+v$.

*You can subtract a Vector3D from a Point3D (giving a Point3D), or from a Vector3D (giving a Vector3D) and you can subtract a Point3D from another Point3d, giving a Vector3D.  BUt Vector3D minus Point3D was deemed meaningless, even though you could decide that it was the negative of the point minus the vector.  The problem with that notion is that it implies multiplying a point by a scalar ($-1)$.

*You can multiply a vector by a scalar, but multiplying a Point3D by a scalar is meaningless.
A: Indeed, two vectors with the same coordinates (represented in the same basis, mind you) is in fact the same vector, since the vector is uniquely given by its coordinates (and its basis). 
Sometimes, when doing calculations, it can be useful to operate with the convention that, unless otherwise stated, a vector has its "starting point" at the origin, so that its coordinates and the point it points to coincide... but this is just one representation of the vector, which doesn't change the fact that it isn't "bound" to the point in space that happens to coincide with the coordinates of the vector.
In fact, vectors are much more general objects than just "arrows with a length"; instead they are elements in a set called a vector space, for which certain operational rules apply. A function, say $\sin(x)$, can for instance be regarded as an infinite-dimensional vector! Look forward to learning about linear algebra, it is quite the ride! 
A: Maybe it could help you to explain the subject as I learned it, that is


*

*two points $A$ and $B$ individuate a (line) segment between them:
$AB$

*an ordered couple of points $(A,B)$ individuate an oriented
segment: $\mathop {AB}\limits^ \to  $

*introducing a reference system, a point corresponds to the oriented
segment from the origin:$A \equiv \mathop {OA}\limits^ \to   $

*a vector is the class (can we imagine it as a  set ?) of all the
oriented segments, whose difference between the coordinates
(arrow-tail) provides the same ordered  n-uple: the coordinates of
the vector.

*a point $A$ and a vector (an applicated vector in engineering and
physics) individuate a point: the $B$ in the $\mathop {AB}\limits^
   \to  $ representing the vector.
So the point $A$ corresponds to $\mathop {OA}\limits^ \to $ which is a representative of the vector having the same coordinates as $A$.
Note: I do not pretend to have exposed the matter in rigorous terms, but the picture should be correct (or I shall with you revise my knowledge basis)
A: The direct answer is "Yes, they are the same vector."
A vector is determined entirely by its direction and magnitude. Location doesn't matter.
A: Vector actually means ordered tuple. So points can be represented as vectors. But displacement between two points will also be vector. And this vector though it looks like point representation vector, is different from Point representation in some applications
