Are points and vectors (in $\mathbb{R}^n$) different objects? If yes, then why can we switch between them in a proof?

Question

Context

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In Hubbard and Hubbard's book on vector calculus, $\mathbb{R}^{n}$ is defined as the space of ordered lists of $n$ real numbers. The authors then say that a given element of $\mathbb{R}^{n}$ can be interpreted in the following two ways:

1. An element of $\mathbb{R}^{n}$ is said to be a "point" (in $\mathbb{R}^{n}$) if it represents some sort of position/state.
2. An element of $\mathbb{R}^{n}$ is said to be a "vector" (in $\mathbb{R}^{n}$) if it represents some sort of change/increment.

Then the book pauses to emphasize that two points cannot be added $(\text{New York}+\text{Boston}=???),$ but two vectors can. It also goes on to define scalar multiplication for vectors, difference of two points, sum of a point and a vector and sum of two vectors.

The authors give the following remark:

"An element of $\mathbb{R}^{n}$ is an ordered list of numbers whether it is interpreted as a point or as a vector. But we have very different images of points and vectors, and we hope that sharing them with you explicitly will help you build a sound intuition. In linear algebra, you should think of elements of $\mathbb{R}^{n}$ as vectors. However, differential calculus is all about increments to points. It is because the increments are vectors that linear algebra is a prerequisite for multivariate calculus: it provides the right language and tools for discussing these increments."

Another important remark is also given:

"Sometimes, often at a key point in a proof, we will suddenly start thinking of points as vectors, or vice versa."

Reflections on H+H's Explanation

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It seems like Hubbard and Hubbard are suggesting that points and vectors are nothing more than labels we give to the same object (a real $n$-tuple). We choose which label we want to give a $n$-tuple based on what we are using the $n$-tuple for/how we are thinking about it.

In this case, points and vectors are the same mathematical object ($n$-tuple), and the only reason for why we can't scale points, or add points to other points is because it breaks our "mental model/interpretation" of points being $n$-tuples that represent locations/states ($\text{New York+Boston=???, and 5}\cdot \text{New York=???}$).

Also, points and vectors being the same objects means there is no harm in switching between the two (provided we don't switch one of them in a way that breaks our interpretation of points/vectors. E.g., if we have $x+y$, for vectors $x$ and $y$, we can switch $x$ with a point and leave $y$ alone (or vice versa), but we cannot switch both $x$ and $y$ out for points because we can't add points). This justifies the second remark from the book.

This all makes sense to me.

• Points and vectors in $\mathbb{R}^{n}$ are the same objects ($n$-tuples), and we just use these two different terms to provide additional context to how we are thinking about/visualizing/using the $n$-tuple.
• Points represent locations/states, vectors represent changes/increments.
• The "rules" of not being able to add two points and not being scale a point are solely there to ensure that our interpretation of what the $n$-tuple is representing is consistent with our intuition $(\text{New York + Boston=???, 5}\cdot \text{Boston=???}).$

But I have run into a few problems.

Questions

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In some of the popular threads of similar questions it seems like some people are claiming that points and vectors in $\mathbb{R}^{n}$ actually ARE different mathematical objects (See What is the difference between a point and a vector?).

That's not what I thought at first reading H+H, but ok, it seems reasonable that points and vectors are different mathematical objects, rather than just contextual indicators for the same object. They are used to represent different things (location vs displacement) after all, and they have different operations that can be performed on them. So I have the following question.

Question 1: Are points and vectors (in $\mathbb{R}^{n}$) simply different interpretations of the same mathematical object ($n$-tuple), or are they fundamentally different objects? If they are different, how are each of them defined?

And if points and vectors ARE different mathematical objects, then how can we reconcile this with the fact that we want to be able to switch between points and vectors in the middle of proofs (Remark 2 from H+H)? This is the second question I have.

Question 2: If points and vectors in $\mathbb{R}^{n}$ are different mathematical objects, how are we able to switch between these two distinct objects in the middle of a proof and still have our proof be valid (i.e., How do we reconcile the fact they are different with remark $2$ from the book)?

Any help at all would be extremely appreciated!

Answer 1

As a mathematical object they are not different. These are only two different interpretations. You can add two elements of $\mathbb{R}^n$ together but when you interprete them as points you don't really know how to imagine that. I think in pure mathrmatics you don't really destinct between the two interpretations, in physics the "difference" is more important. In my courses the elements of $\mathbb{R}^n$ were formally defined as functions $f:\{1,...,n\}\rightarrow\mathbb{R}$.

Answer 2

To Question 1

The collection of points in real 3D space (i call it $E$) is (assumed to be) a manifold. We can identify any point $p$ in $E$ with a vector in $\mathbb{R}^3$ (a triplet of numbers) in the following way:

• Choose a point of $E$ as the origin $o$ of $E$
• Choose three other points $e_1, e_2, e_3$ of $E$ so that the arrows $\vec{oe}_1, \vec{oe}_2, \vec{oe}_3$ pointing from $o$ to them are linearly independant.
• Now we can identify $p$ with the arrow $\vec{op}$ that points from $o$ to $p$ and uniquely decompose this arrow in terms of the arrows $\vec{oe}_1, \vec{oe}_2, \vec{oe}_3$ in the form $\vec{op} = \sum_{i=1}^3 \alpha_i \vec{oe}_i$, where the $\alpha_i$ are real numbers
• Now we identify $p$ with the triplet of numbers $(\alpha_1 ,\alpha_2, \alpha_3) \in \mathbb{R}^3$ (these numbers are called coordinates of $p$)

Note that there is no natural way of identifying a point in $E$ with a triplet of numbers, there are many different ways of doing so.

There is no natural way to add points from $E$ (like New York + Boston= ?), but we can add the coordinates of two points in $E$. For example we can add the coordinates $(N_1, N_2, N_3)$ of New York to the coordinates of Boston $(B_1, B_2,B_3)$ to obtain a triplet of numbers $(C_1, C_2, C_3)= (B_1+N_1, B_2+ N_2, B_3 + N_3)$.

From this we obtain an arrow $\sum_{i=1}^3 C_i \vec{oe_i}$ that points from $o$ to some other point in space.

Note that this arrow points to where we would end up if we would go from $o$ to New York and then go from New York in the same direction as the arrow from $o$ to Boston points (for example south) for the length of that arrow.

This is a physical axiomatic description of 3D space (and not a mathematical one). It is not as fundamental as the vector space $\mathbb{R}^3$, which exists even if $E$ does not. In some sense points of 3D space are not mathematical objects but rather physical objects whereas triplets of numbers $(\mathbb{R}^3)$ are mathematical objects that we can use to describe these physical objects.

This way of identifying 3D points with numbers (and vice versa) is so natural to us humans that we do not even think about it when we for example plot a function or some surface.

Now consider $\mathbb{R}^n$ as a set of points (and a manifold). Then for any $p \in \mathbb{R}^n$ we have the tangent plane $T_p \mathbb{R}^n$ to $p$, which we can imagine as arrows pointing from $p$ to some other point (the displacements). It is a standard result that $T_p\mathbb{R}^n$ is isomorph to $\mathbb{R}^n$ itself. If you dont understand what this means/want to learn more see any textbook on smooth manifolds. Then $\mathbb{R}^n$ (as the points) is not the same as $T_p \mathbb{R}^n$ (the displacements) but the two can be identified in a natural way. This analogy of points and displacements essentially bases on the notion of 3D-space established earlier.

To Question 2

When we switch between the two views (as points of $E$ and as triplets of numbers) we are essentially basing our proof on physical axioms (or axioms of euclidean geometry) and manifold theory. This does change the validity of these proofs. But usually in applied scenarios like physics or engineering these axioms are assumed.

As a consequence in pure maths it is usually tried to stay completely in the $\mathbb{R}^n$ framework, because the construction of $\mathbb{R}^n$ uses far less axioms than euclidean geometry/physics does. In many (maths) manifold textbooks you will not find a single mention of the 3D space of points $E$.

Answer 3

The point they are making is a subtle one. You can think of points and vectors as the same thing, and this is fine. But, to elaborate on their distinction a little, the set $\mathbb{R}^n$ is an example of many types of mathematical object. It is a set of course. It is also a vector space (hence the elements are vectors). It is also other things, like it is a group. It is a manifold/geometric object, too. The elements of $\mathbb{R}^n$ are just elements, but when we are thinking of them as elements of a vector space, we might think of them as vectors. When we are thinking of them as elements of a manifold, we think of them as points. And so on...