# Why do vectors and points use similar notations?

Alright a vector is suppose to be magnitude with direction or rather in a simple sense, "numbers with direction".

What really gets to me is that why would a point and a vector share similar notation for describing themselves ?

Lets take a point in a 3D space $(2,3,4)$, and then lets take a vector $[2,3,4]$.

Why are they using almost the same notations for describing two different mathematical concepts ? Also how is the direction being denoted in the this notation ? Is the direction from the origin ? Can't that be applied to the point as well ?

I am really confused about this. To be clear I have already read this question, but my question is why use similar notations in both cases, and also how the direction information is inferred from the present notation ?

• There's a one-to-one correspondence betweens vectors and points. So usually we just conflate the two concepts. If you'd like to keep them separate, look up the concept of an affine space or wait until you get to the concept of a manifold in differential geometry where you're forced to distinguish them. – user137731 Jun 8 '17 at 18:38
• When discussing the direction of a point (though usually we'd use the word vector if we care about direction), we mean the direction from the origin to that point. – user137731 Jun 8 '17 at 18:40
• They're all vectors. Some vectors are displacement vectors, indicating a general movement in a certain direction by a certain distance and some vectors are position vectors indicating a movement starting from a 'special' location (the origin). – Paul Aljabar Jun 8 '17 at 18:42
• @ng.newbie Actually, looking at the question you linked to, Robert Israel has it down pat: we distinguish the two concepts when it's convenient and don't when it's not necessary. – user137731 Jun 8 '17 at 18:43
• @PaulAljabar I've never liked that position. If you're going to distinguish them it's better to just introduce the concept of an affine space so you can do so without all the hand-waving. – user137731 Jun 8 '17 at 18:44

## 3 Answers

We call them vectors because they belong to vector spaces. This has the 'context baggage' of meaning we can add two of them together and multiply them by a constant (and these operations behave as you would expect from the arithmethic analogues).

However, one key ingredient of vector spaces is a set (the other being the aformentioned operations). As such, elements of a vector space are not just vectors; they are first and foremost points (of the set).

Visualizing them as points or as 'arrows' is really a matter of context and convenience.

The mathematical notion of vector is much larger than what is initially presented. To a mathematician, you start with an algebraic system $F$ of numbers where you are allowed to add, subtract, multiply and divide as usual. Usual examples are to take $F$ to be the real numbers or the rational numbers or the complex numbers (but not, for example, the integers because you can't divide: $3/2$ is not an integer). Then a vector space is any algebraic system of vectors where you can add two vectors and scalar multiply by elements of $F$.

Examples:

I will let $F$ be the real numbers in all of these examples.

• real numbers where $a + b$ and $a\cdot b$ are just the usual addition and multiplication

• pairs of real numbers where $(a,b) + (c,d) = (a + c, b + d)$ and $k(a,b) = (ka, kb)$

• more generally: $n$-tuples $(a_1,\dots,a_n)$ of real numbers where addition and scalar multiplication are defined by $$(a_1,\dots,a_n) + (b_1,\dots,b_n) = (a_1+b_1,\dots,a_n+b_n) \text{ and } k(a_1,\dots,a_n) = (ka_1,\dots,ka_n)$$ we call this $\mathbf{R}^n$ or $n$-dimensional euclidean space.

• an arrow from the point $(0,0,\dots,0)$ to $(a_1,\dots,a_n)$. The addition is done via the parallelogram law and scalar multiplication is done by stretching the arrow. If we denote this arrow by $[a_1,\dots,a_n]$ then we see that $$[a_1,\dots,a_n] + [b_1,\dots,b_n] = [a_1+b_1,\dots,a_n+b_n] \text{ and } k[a_1,\dots,a_n] = [ka_1,\dots,ka_n]$$ so really we didn't do anything other than change $(, )$ to $[, ]$ and started calling these arrows rather than points.

• a weirder one: polynomials are vectors because we can add polynomials and multiply polynomials by real numbers. If we look at all polynomials of degree less than $2$ (i.e. constant, linear or quadratic) then they all look like $a + bx + cx^2$ with $a, b, c$ real numbers. We can then associate this to a tuple $(a, b, c)$ and then addition and scalar multiplication of such polynomials is done as we did before with $n$-tuples.

The reason we think of these all as vectors, and use similar notation, is to emphasize the similar algebraic properties (being able to add and scalar multiply). In this way, we move away from thinking about concrete vectors and move onto an abstract notion of vector (this is where the terminology "abstract algebra" and "abstract linear algebra" comes from).

This is useful if you want to study the theory of vectors but if you need to do computations it is also helpful to work with concrete vectors. In applications, the difference between an arrow and a point is very important. For actually adding vectors together, it doesn't make a difference.

• So tell me what the meaning when I say that I am adding two vectors ? Also why isn't it possible to add 2 points ? And correct me if I am wrong, but we call those tuples, vectors, where we can use the algebraic properties of addition and multiplication right ? Also given a tuple that is a vector how is the direction inferred from that ? is it the direction from the origin ? – ng.newbie Jun 9 '17 at 11:55
• @ng.newbie Adding two vectors is just adding two vectors. The meaning changes depending on the application. For instance, if your vectors are real numbers then adding them is just adding real numbers. This might for instance correspond to a particle travelling on a line. We can add points by adding their coordinates. This can be interpreted in several ways. For instance maybe you have a warehouse with 3 couches and 4 chairs and you get a shipment in of 1 couch and 2 chairs. This is represented by the calculation $(3,4) + (1,2) = (4,6)$. Maybe multiplication doesn't make sense, but we can add. – Trevor Gunn Jun 9 '17 at 13:12
• @ng.newbie The point of working with tuples is that they expose the key numerical information and remove the physical meaning. This is exceptionally useful, for example, on a computer, where vectors can be stored as lists or arrays of numbers. You can think of a tuple as being an arrow pointing out of the origin to a given point but this is physical information. When you give all of this to a computer, your computer isn't thinking of the vector as an arrow, it is storing it as a list of numbers. – Trevor Gunn Jun 9 '17 at 13:16
• So even real numbers can be vectors ? Then what are scalars ? I thought vectors were real numbers with direction not just real numbers. – ng.newbie Jun 11 '17 at 12:15
• @ng.newbie Vectors are anything you can add together and multiply by real numbers. So this includes real numbers. "Direction" is part of the physical interpretation of vectors, it's not an algebraic property. The only difference between a "real number" and a "real number with direction" is how you think about it. The algebra (adding and scalar multiplying) remains unchanged. – Trevor Gunn Jun 11 '17 at 14:54

In fact, vectors are a far more abstract concept. A vector is an element of the set of a vector space, which again is a set with some operators that have some specific properties. It shows, that the set of n-tuple of real numbers with addition and scalar multiplication has those properties. And, more important, that all finit-dimensional vector spaces are isomorph to this space (you could say, that "they're the same").

You are actually adressing a more physical interpretation problem. You can interprete these n-tuples either as points or as directions (which I assume you mean when you're talking about "vectors"). Actually it isn't a real problem, because you can switch between the interpretations whenever you want. Let me give you an example.

Conside space curves: $c: [a,b] \rightarrow \mathbb{R}^3$ and the corresponding tangential vectors $T: [a,b] \rightarrow \mathbb{R}^3, t \mapsto \frac{c'(t)}{|c'(t)|}$. Both are functions from a real intervall to the 3dim-space. But in the first case, you interpret the vectors as points in the space, and the range is what you would think of a curve. However, the tangential vector is interpreted as the direction of the curve in every point. Usually you "glue" this vector to the corresponding point of the curve, but you can also let them be vectors in 0; then you get only a circle arc.

However, there are situation where you need both, an anchor point and a directions. In those cases you just consider two vectors: The first one describes the anchor point, the second one the direction. If you are interested in this topic, see tangential bundles.