Vectors vs. Ordered Pairs

Question

I am currently learning calculus and am still a little confused as to the difference between vectors and points (which are represented as ordered pairs). I know that vectors are a different type of object given that they have both direction and magnitude, but I don't understand why they are inherently different given that they do not seem to communicate any more information than a point represented as an ordered pair does.

Furthermore, is there something "special" I have to do to convert an ordered pair $(a,b)$ into a vector $<a,b>$ or not?

Answer 1

Ordered pairs can be vectors, but they are not the same. It's not a proper definition, but everything is a vector that can be multiplied by a scalar and they can be added together. For example vectors, numbers, functions. For a mathematically more correct definition see the Vector Space.

In Linear algebra, we were told something like this: Let $P$ be a point in the space. If you choose an origin, then you can have a vector $\vec v$ pointing to $P$. After choosing a base, you can represent the vector $\vec v$ as an unique $n$-tuple $\underline{v}=(v_1,v_2,\dots,v_n)$.

Answer 2

A vector is an object on its own right. To represent a vector as an ordered pair, you need to fix a basis. If you chose another basis, the same vector will be represented by a different pair. And strictly speaking you don't need any basis at all to observe various relations between vectors.

Those various pairs representing the same vector are not arbitrary however; they are related via the transformation from one basis to another. If some ordered pair does not obey the transformation rules, it does not represent a vector.

Answer 3

I understand your confusion and I'm going to try help you a bit.

First: Vectors are the elements of a vector space. I don't know if you know the (abstract) definition of a vector space. Anyway, vectors are much more than points or arrows. For example, real numbers can be considered themselves as a vectors. And the same apply for pairs $(x,y)$, triples $(x,y,z)$ and so on. Continous functions defined on an interval $[a,b]$ is another example of vector space; here the vectors are the functions. So, as you can see, vectors are a very rich topic.

Second: If your vector space is real (complex) and finite-dimensional then, you can study it as the vector space $\mathbb R^n$ (resp. $\mathbb C^n$), for some appropiate $n$. So, at the end of the day, you have points.

Third: Probably your confusion cames from the structure of affine spaces. Roughly speaking, an affine space is a vector space $V$ together with a pair $(P,g)$, where $P$ is a set (the set of points) and $g:P\times P\rightarrow V$ is a map that assigns a vector to any pair of points $p,q$, (with some rules).

Now, on this situation, imagine yo have a preferred point $\mathcal O$ on $P$, which we will call the origin. Then, for every point $p\in P$, you have defined a canonical vector on $V$, namely the vector

$$ g(\mathcal O, p) \equiv \vec{\mathcal O p}\equiv \vec p . $$

This vector has an application point ($\mathcal O$), direction and length (magnitude) and is different than the point $p$.

So, answering one of your questions yes, for any pair of points you have a way to define a vector: the map $g$.

Answer 4

The vector space $\mathbb{R}^2$ is the same as the points in the plane. Whether to consider a vector a point or a point a vector is all about your point of view and what you are trying to accomplish.

Answer 5

Vectors come from physics, where they are represented by directed line segments -- line segments with arrowheads. Thus, vectors and points are different. First, because points have no specific direction and no magnitude of any sort even.

Concerning the algebraic notation, points are represented by ordered tuples when a coordinate system has been fixed -- there's nothing more to it -- an ordered list of real numbers. Vectors may also be represented by such tuples, but only as a manner of writing, as a shorthand; for although the notation is formally similar to that of points, they represent different objects. Thus when a coordinate system has been fixed, and the corresponding basis vectors have been chosen, then every vector in that space may be represented as some linear combination of the basis vectors -- we compress this notation further by simply writing the coefficients of the basis vectors that sum to give that vector (but the basis vectors are always implicitly assumed or kept in mind).

Although every point in some space may be related to some vector, what that gives you is a vector-valued function, not an identification of vectors and points. Also, the tip of vectors may be identified with some point, but this is not a fitting correspondence because the same vector may map to many points -- indeed infinitely many. This is as a result of how vectors are defined -- we care only about their magnitude and direction, not their origin or endpoint. Thus two parallel vectors with equal magnitude (that are not coincident) are the same, even though their tips correspond to different numbers.