Confusion about the usage of points vs. vectors

As far as definitions go, understand the difference between a vector and a point. A vector can be translated and still be the same vector, whereas a point is fixed. But I would like some clarification on the usage of vectors and the usage of points, because it seems like in many cases they are used interchangeably. For example:

1. It has always been my understanding that addition is not defined for two points. But in this question, two points are being added together in this equation:

$\overline{C}= \lbrace tx+(1-t)x': x\in C, x' \in C', t\in [0,1]\rbrace$

2. Sometimes $\mathbb{R}^n$ is used to denote the set of $n$ dimensional vectors, and sometimes it denotes the set of points in $n$ dimensional space.
3. In vector calculus, it is often said that a function with multiple inputs takes a vector as an input, but I have rarely seen a function written as $f(\vec{v})$. Even though I understand that what it means to say is, the vector that originates at the origin, to me, it doesn't seem entirely correct to say that the input is a vector without explicitly saying that the vector's tail is at the origin.

Can anyone clarify these points of confusion? Are point and vector interchangeable in these cases? Also, is there a notation for "converting" one to the other? E.g. how to "convert" $\langle{x,y,z}\rangle$ to $(x,y,z)$ or vice versa?

I'll try to convince you that they are geometrically quite obviously different, but when it comes to naming them, the begin to look alike :)

Geometrically you probably already have a good picture of what a point is: it's just the primitive notion of a point you have in geometry. That is, a single dimensionless location in space.

A vector should be thought of as having two qualities: a ray that has direction and magnitude. In basic vector algebra in $\Bbb R^n$, we learn that such a ray can slide all around $\Bbb R^n$, and as long as you aren't changing the direction or the length of the ray, then it is still the same vector.

Now when it comes to naming these two things, they start to look alike! With Cartesian coordinates, points in $\Bbb R^n$ are labeled by their projections to the axes, and that creates a list of real numbers. Similarly, when we go about naming vectors, we have this convention of sliding the vector so that it is being emitted from the origin, and then we check to see what point is on its arrowhead. The vector is named after this point.

So in both cases, a similar list of real numbers is used to identify the object. Since this is the case, it's common to just start referring to any ordered $n$-tuple of things from a field (like $\Bbb R$) as a "vector," even if we aren't thinking of it as a ray in that application.

One example is that of vector fields. Since these are functions of position, the inputs they take are points of $\Bbb R^n$ (which look like an ordered $n$-tuple). The outputs are vectors (which again look like an ordered $n$-tuple), but we are interpreting these as the vectors they represent, slid over from the origin to the point we're at.

You can, of course, really have vector inputs! For instance, the length of a vector in $\Bbb R^n$ creates a function from vectors into $\Bbb R$. Of course, the same function could be reinterpreted as the distance-to-zero function on points of $\Bbb R^n$.

So, the difference is all in how you are interpreting that particular list of numbers.

For #1 in your post, you are probably thinking of it as the line segment between points $x$ and $x'$. The addition that's going on is vector addition though. Drawing the vectors that $x$ and $x'$ represent, you see you have two vectors extending from the orign to these two points. For any two vectors $v,w$, $v-w$ yields the vector which fits between the two tips of $w$ and $v$, and points to the tip of $v$. So, you can see that $x-x'$ has the point $x$ on its tip.

What does the $t$ contribute? If you multiply out $xt+(1-t)x'=x'+t(x-x')$, you can see that the vector $x-x'$ is being scaled by $t$ to something shorter, and then is being concatenated onto the tip of $x'$. The tip of this arrow gives another point on the segment. Ranging over all $t$ between 0 and 1, you get vectors pointing to all points on that segment.

1. It is indeed true that addition is not defined for points. But subtraction can be defined (for flat spaces, at least), and then gives a vector: If $A$ and $B$ are points, then $v=B-A$ is the vector pointing from $A$ to $B$. Also you then can add a vector to a point, giving another point (namely the point where the vector ends if you start it at the given point). So for the vector $v$ from above, $A+v=B$.

Now, the points in the straight line segment from $A$ to $B$ have the form $A+t(B-A)$ with $t\in[0,1]$. Now if you ignore that you're dealing with different types and apply the normal arithmetic laws, you can expand $t(B-A)=tB-tA$ and then factor $A-tA=(1-t)A$. As written, it doesn't make sense with points and vectors as defined above.

However there's also another way to see it: You can identify each point with the vector fro the origin to that point. As soon as you do that, you've got only vectors, and thus can apply all vector operations. The expression you've seen does make immediate sense in that case, because you've basically dispensed with points.

Finally, there's yet another way to think about it, which combines the advantages of both: Add an additional number to the tuples, which has the following meaning:

• If it is 1, you've got a point.
• If it is 0, you've got a vector.
• Any other value is invalid.

Then you can allow any operation where you get a valid (i.e. vector or point) result. For example, in 2 dimensions, you'd have $A=(a_1,a_2,1)$ (the final $1$ marking it as point), and $B=(b_1,b_2,1)$ (again, a point). Now $A+B$ would not be allowed (last component would be $2$), but $A-B$ is allowed and gives a vector (last component $0$). And the formula in question gives as last component $(1-t)+t=1$, so it is valid and a vector.

Note however that all those things only work on flat spaces with linear coordinates. On non-flat spaces or general coordinates you'll not be able to do such an identification (or even to define the vector from one point to another).

2. $\mathbb{R}^n$ is the set of n-tuples of real numbers, not more and not less. Now together with an operation $+:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n$ and an operation $\cdot:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$ which fulfil certain relations, you get the vector space $(\mathbb{R}^n,+,\cdot)$. If you supply other operations, you get something different. Now it is quite common to only write the set when you mean the complete algebraic structure, meaning that the rest of the structure is implied. So if you speak about the vector space $\mathbb{R}^n$ you actually mean "the vector space you get if you use $\mathbb{R}^n$ together with the canonical choice of component-wise addition and scalar multiplication". If you speak about the points as $\mathbb{R}^n$, you of course normally don't consider the vector space structure included.

3. A function may operate on vectors, on points, or simply on $n-tuples$. All those concepts make sense. A function on points could e.g. be a temperature map: Each point is mapped to the temperature at that point. Here, it would not make sense to consider it as function of a vector (except if identifying points with vectors, see above). A function on vectors could e.g. be the function which tells you the length of the vector. That function makes sense only on vectors, not on points. And of course any function on $n$ arguments operates on $n-tuples$. If you describe your points or vectors with $n-tuples$, you'll give the function on those argument as $n-tuples$ as well. On the reverse, since derivations are linear, and linear spaces are vector spaces (indeed, both are just two names for the same thing), whenever you do derivations on a multi-argument function, you can interpret that argument tuple as a vector.

You can "convert" a point to a vector by subtracting the origin (which is a point), and you can "convert" a vector to a point by adding it to the origin.

The most important difference between a vector and a point is that vectors can be added, while points cannot. Instead, you can take two points and "subtract" them to get the vector between them, and add a point to a vector and get a new point. There's really no notion of a vector's "tail": in some sense, a vector just represents the difference between it's head and tail.

When we talk about a (real) vector, we usually mean an element of $\Bbb{R}^n$, which we give componentwise addition and multiplication. A point is an element of an affine space over $\Bbb{R}^n$. However, if we impose coordinates on that affine space, which is equivalent to picking a single point of the affine space and calling it the origin, we get a correspondence between points and vectors. So in your question 1, those points can be added, because they are in a coordinate system, and. Similarly, in question 2, since affine spaces are so closely related to vector spaces, we can use the same notation for both.

Regarding your question three, the notation $\vec{v}$ is really just helpful to avoid confusion, but, in most cases where confusion between vectors and scalars is uncommon, is unnecessary.

Note both that a vector can be defined more generally as an element of a vector space, which is a much more general notion than just $\Bbb{R}^n$, and that "point" can refer to elements of a much broader variety of spaces, like topological spaces or affine varieties.

In general points don't have to sit in a vector space, and in that context they become different. But when it comes to points lying in a vector space, I don't see why you couldn't define addition of points in a vector space as addition of the vectors from the origin to those points. If you do so, they are of course isomorphic - the points are in bijection with the vectors, and adding two vectors before turning them into points can't be distinguished from turning two vectors into points and adding them. It's a difference of semantics alone, but it's clearly not a meaningful difference of semantics, and I find it very useful to, for example, think of a polynomial as a set of vectors, its roots and any factor of a quadratic without a real root.