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I understand that a vector has direction and magnitude whereas a point doesn't.

However, the course note that I am using states that a point is the same as a vector.

Also, can you do cross product and dot product using two points instead of two vectors? I don't think so but my roommate insists yes and I'm kind of confused now.

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Physicists will tell you they are different things. Mathematicians will tell you they are the same. In mathematics they are defined as being the same and that's enough. –  Git Gud Jan 21 at 0:57
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@Git Gud: many mathematicians will tell you they are different, I suspect. –  Carl Mummert Jan 21 at 1:03
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An element of $\mathbb R^N$ could be called either a point or a vector, depending on how we are thinking of it/how we're visualizing it/what we're doing with it. Note that both words point and vector also have other meanings in math. –  littleO Jan 21 at 1:25
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@copper.hat: Right, but the OP's question doesn't say "affine space", and the elements of e.g. $\mathbb{R}^n$ are certainly also called "points". When people say "Well, strictly speaking it makes no sense to add points", I feel that it is an unhelpful statement. Of course it makes sense to add points in a structure where addition is defined. There is a way to view Euclidean space -- namely as an affine space -- in which addition of points in not sensible. That is not the same as saying that addition of points is not sensible. –  Pete L. Clark Jan 21 at 2:06
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A point is a pure geometric/topological concept. It is an element from a set which we visualize as some sort of "space". Vectors are points with some extra algebraic assumptions imposed upon them. More precisely, you can add vectors and scale them by a scalar. If you are working with elements in $\mathbb{R}$ whose points are really vectors, there are no difference in referring an element of $\mathbb{R}^n$ as a point or as an vector. –  achille hui Jan 21 at 2:26

19 Answers 19

What exactly is a vector? You are right that we usually consider a vector as something that has a direction and a magnitude, but there more precise and abstract definition is that a vector in, for example, $\mathbb{R}^n$ is just an element of that set. So it is the same as a point when you consider it as an element of a set.

Now if you want to talk about cross products and magnitudes, then it becomes a question about linguistics. The way you, for example, define the magnitude as the function $$ \lvert\cdot\rvert: \mathbb{R}^2 \to \mathbb{R} $$ given for $a = (a_1, a_2) \in \mathbb{R}^2$ by $$ \lvert a \rvert = \sqrt{a_1^2 + a_2^2}. $$ So if you insist on talking about the magnitude of a point, then you are "free" to do so (i.e. free to define this). But bare in mind that you will also cause confusion by doing this. And with doing math, we want to communicate clearly and so ...

In the same way, you could define the addition or cross product of points.


Maybe it would be better to say this: Is the vector space the same as a set? Yes, a vector space is a set. But it is also more than a set. We can't add elements of a set, but we can add elements of a vector space because with a vector space you get the definition of an addition. So in this sense, a point and vector are very much different.


Added: If you want to find the equation of a plane that contains the three points $a$, $b$, and $c$, then you would not subtract the points. So how do you so it. Well, if the coordinates to point $a$ are $(a_1, a_2, a_3)$, i.e. if $a = (a_1, a_2, a_3)$, (and likewise for $b$ and $c$) then you first define the vectors $$ \vec{ab} = (b_1 - a_1, b_2- a_2, b_3 - a_3) $$ and $$ \vec{ac} = (c_1 - a_1, c_2- a_2, c_3 - a_3). $$ Then a normal vector for/to the plane is the cross product of the vectors: $\vec{ab}\times \vec{ac}$.

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So can you cross two points using cross product? –  6609081 Jan 21 at 0:58
    
@6609081 Yes: what you're doing, if you want to think in terms of vectors, is taking the cross product of the vector pointing from the origin to the first point with the vector pointing from the origin to the second point. –  Omnomnomnom Jan 21 at 1:00
    
@6609081: you cannot literally take the cross product of two points in $\mathbb{R}^3$. But sometimes we treat points "as if" they are vectors, –  Carl Mummert Jan 21 at 1:01
    
And that's why I'm confused about whether you can cross two points or not. Cause if you can directly cross two points, why would you first do the subtraction? –  6609081 Jan 21 at 1:08
    
@6609081: As I mention in my answer, I would not talk about taking the cross product of points. But you can make a sense of this. –  Thomas Jan 21 at 1:11

In spirit they are different things. But the usual convention is to think of vector in the plane or in three-dimensional space as starting at the origin. In that case, a vector is identified precisely by its ending point, giving you an identification between points and vectors.

One way to see that they are different things (even if identified in many circumstances), is that you can add vectors, while the sum of points makes no sense. Same with the dot and cross products.

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It is not usual to think of a vector in the plane...as starting at the origin. This is how they are introduced, yes, but for any decent application this doesn't work. For example, you could not talk about vector fields if all of the vectors started at the origin! –  user1729 Jan 21 at 16:44
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Yet, the only algebra you can do on a vector field is pointwise operations, which means you are considering the vectors as starting at the origin. Of course one can think of putting vectors anywhere, but the time when then become useful is when you can operate with them, as that's done when they share their starting point, which is the same as saying that they start at the origin. –  Martin Argerami Jan 21 at 20:23
    
What you are saying is subtly different from thinking of it as beginning at the origin. Your point, I believe, is that you don't have to worry about them not starting from the origin if you want to perform an operation. However, in many ways the point of vectors is that they can start anywhere! Without this, they loose their power. –  user1729 Jan 22 at 11:50

Points and vectors are not the same thing. Given two points in 3D space, we can make a vector from the first point to the second. And, given a vector and a point, we can start at the point and "follow" the vector to get another point.

There is a nice fact, however: the points in 3D space (or $\mathbb{R}^n$, more generally) are in a very nice correspondence with the vectors that start at the point $(0,0,0)$. Essentially, the idea is that we can represent the vector with its ending point, and no information is lost. This is sometimes called putting the vector in "standard position".

For a course like vector calculus, it is important to keep a good distinction between points and vectors. Points correspond to vectors that start at the origin, but we may need vectors that start at other points.

For example, given three points $A$, $B$, and $C$ in 3D space, we may want to find the equation of the plane that spans them, If we just knew the normal vector $\vec n$ of the plane, we could write the equation directly as $\vec n \cdot (x,y,z) = \vec n \cdot A$. So we need to find that normal $\vec n$. To do that, we compute the cross product of the vectors $\vec {AB}$ and $\vec{AC}$. If we computed the cross product of $A$ and $C$ instead (pretending they are vectors in standard position), we could not get the right normal vector.

For example, if $A = (1,0,0)$, $B = (0,1,0)$, and $C = (0,0,1)$, the normal vector of the corresponding plane would not be parallel to any coordinate axis. But if we take any two of $A$, $B$, and $C$ and compute a cross product, we will get a vector parallel to one of the coordinate axes.

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Would you address the philosophical difference about what the "essence" of a mathematical object is? I am guessing that you hold that the essence of an object is not just given by the element that it is in the set, but also by what we can do with the element. This might be better as a separate question, but would you say that by starting with a set and then defining a function from the set to, for example, the real numbers, we have changed the essence (I don't know what else to call it) of the elements of the set? –  Thomas Jan 21 at 2:00
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@Thomas, I think I see what you are saying, but I am not sure how to answer. My own view is that it's important to think in terms of "types". Sometimes we can convert an object between different types; a point can be converted to a vector from the origin, and vice versa. But each function is really defined on a certain type, and keeping them straight is an important part of doing mathematics. –  Carl Mummert Jan 21 at 2:12
    
The points in 3D Euclidian space can themselves be viewed as vectors. The abstraction of the eight axioms for a vector space can be applied to 3D space by defining addition to be coordinate wise. This addition comes up with exactly the same results that using arrows as geometric objects manipulated with some trigonometry or whatever. Okay the points are referenced to an origin and in physics we have to deal with relativity, but mathematical abstraction often leads to better ways of doing the same thing. –  Geoff Pointer Jan 21 at 5:13
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Good explanation, but I just want to point out that the wording "vectors that start at other points" can be misleading; vectors do not have starting points. I'm not sure how to say it better, though. Perhaps "while a point can be represented by the vector between that point and the origin, a vector can also define the separation (difference) between any two points, not just a point and the origin." That still seems a bit hard to digest, though. –  Dr. Wily's Apprentice Jan 21 at 19:50
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@Dr. Wily's Apprentice: I hear what you are saying. This is one of those difficult issues. On one hand (especially in elementary calculus) we would think of a vector as representing a distance and displacement, not attached to any point. But later we would think of a vector as an element of a tangent space at a particular point. In that sense every vector does have a distinguished starting point. (And, of course, we find the tangent plane to a surface at a point using two vectors parallel to the surface at that point...). This is why I put "follow" in quotes - it could be parallel transport. –  Carl Mummert Jan 21 at 20:21

Well, there is a huge technical difference. Although, the conceptual difference really depends on how one tries to visualize things, and that for which we need to visualize them.

In some subjects, such as calculus, I generally imagine points and vector ad libidum (dots and arrows floating around graphs of functions). This is because there's typically (in beginner calculus) an implicit convention to use standard unit vector coordinates. In geometry though the heavy technical difference almost calls for a conceptual difference. I will try to illustrate the technical difference as follows: points are elements of some set, call it $G$ whose characteristics we don't necessarily know, but if we can always find a unique vector $^*$ that corresponds to two points $p,q$ (in that order), then we have a new structure on $G$ (the set of points) called affine geometry (basically "normal" geometry, except for distances). Of course since we now have this correspondence we can imagine any point $q$ as the unique vector that corresponds to the two points $o,p$, for a certain point $o$ (the origin), but fundamentally points and vectors are two different things. Additionally, in other geometries the interaction of points and vectors may be different.

*Along with all of this, a vector has its own definition which has to do with operations such as adding and multiplying by scalars, and no reference is made to points whatsoever.

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Vectors and points are two different things and should not be confused. They both share certain similarities, which makes the transformation of one into another very easy, but they are used in different ways as well as describe different mathematical objects.

A point is a location in a coordinate system, that is a location defined relatively to an origin. If you were to move the origin without moving the point, then the coordinates of the point would change.

A vector is a more general object. No matter where you draw a vector $\vec{v}$ on a plane, it is still the same. If you were to move the origin, the components of the vector would not change. You can also think of a vector as a transformation. It can be applied anywhere and have the same effect: displacing a point of a certain distance in a precise direction.

The confusion between the vector and the point comes from the fact that a point $P$ may also be represented as a vector $\vec{OP}$, that is the vector starting from the origin $O$ going to the point $P$. Only then are the vector and the point somewhat equivalent. A vector defined as $\vec{AB}$, with $A$ and $B$ points, should not be confused with some point $X$ such that $\vec{AB} = \vec{OX}$ .

You can understand that even though it is sometimes useful to represent a point as a vector, you should usually not represent a vector as a point.

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Indeed. One way to think about it is this. Consider a point in a fixed coordinate system: it cannot be moved and is therefore also static or fixed. A vector, on the other hand, has two properties: a magnitude and a direction. Neither of these say where the vector is relative to the fixed coordinate system. A vector can be moved about wherever you wish so long as the magnitude and direction stay the same. Additional note: these 'objects' (ie. point, vector) can also be distinguished by how they are operated on. The dot & cross products of points have no meaning. For vectors they do. –  zentient Jan 21 at 7:04

This is a question which causes a lot of confusion and it's good that you are trying to clear it up as early as possible. It is clearly a question about the geometric meaning of vectors, so IMHO it is not helpful when people start to involve vector spaces in the discussion.

Let me make the assumption that you know what a point is, and that the confusion begins when vectors are introduced. I don't know how to include diagrams in a post so I must ask you to draw your own. You can visualise a vector as an arrow in the plane (or in $3$-dimensional space, but let's stick with a plane for now). The usual understanding is that a vector is specified by its length and direction, and not by where it is located in the plane. For example, draw an arrow from $(1,-2)$ to $(3,1)$ and another from $(0,2)$ to $(2,5)$. The two arrows have the same length and direction, so they are regarded as the same vector, and it can be written as the vector $(2,5)$, or $2{\bf i}+5{\bf j}$ if that's the notation your instructors use.

We often use language a bit loosely and refer to a point as a vector. (It would be more precise to say the point is represented by the vector, but contrary to popular belief mathematicians are not always 100% accurate in how they speak!) In this case we mean the vector from the origin to the stated point. So the first vector drawn above does not represent the point $(3,1)$ since it does not start from the origin. On the other hand, if you draw the arrow from $(0,0)$ to $(2,5)$ then you can see that it is the same vector (that is, has the same length and direction) as the other two. Since it starts from the origin, this vector represents the point $(2,5)$ - as do the other two, since they are the same vector. As you can see, a vector from the origin and the point it represents are the same numerically, but they are different conceptually and it's worth spending some time trying to get your head around it.

Another example - if you haven't seen this yet I expect you soon will. The equation of a line can be written in "parametric vector form" as, for example, $${\bf x}=(1,2)+\lambda(3,4)\quad\hbox{for $\lambda\in{\Bbb R}$}.$$ Here we think of the vector $(1,2)$ as specifying a point on the line and $(3,4)$ as specifying the direction of the line. So it is important that we should draw $(1,2)$ as starting from the origin (please draw it), but it is not important where we draw $(3,4)$, and the easiest way is to draw it starting at $(1,2)$ and going to $(4,6)$. Then you can draw in the line through $(1,2)$ in the direction $(3,4)$, and this is the line specified by the above equation. The notation ${\bf x}$ will be a variable point on the line, or in other words a variable vector from the origin to the line.

Hope this helps - good luck!

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Point usually refers to topological structure. Eg., point-set topology; a sequence of points converging; a neighborhood of a point; etc.

Vector usually refers to vector space/normed space/inner product space structure - Eg., adding two vectors; scaling a vector; taking the norm of a vector; etc.

In a set with both structures - a vector space with a specified topology - the context of the argument tends to determine which word is used.

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Points describe locations while vectors describe directions.

Imagine a ball. A series of coordinates can be used to describe the points on this ball. Similarly, at any given point, a series of components can be used to describe directions (vectors) in the vicinity of a point.

Both use a collection of numbers to describe, so it may not be obvious how they are different, but vectors can be added or subtracted, multiplied by scalars, and so on. The vector space structure is essential to the algebra of directions: you can add any two directions to get a third, for instance.

Is it meaningful to add points? You can add coordinates of points, and you may or may not end up with another point. It's not obvious that two different ways of assigning coordinates to points would allow points to be "added" in a way that gives the same result for both systems--in fact, I do not believe this to be true. Consider, for instance, what happens if you have the point $(\theta, \phi) = (\pi/2, \pi)$ on a sphere and added the point $(\pi/2, -\pi)$. You would get $(\pi, 0)$, which describes the south pole. But, it's equally valid to describe the top hemisphere in terms of $(x,y)$, and you'd get $(0, 1) + (0, -1) = (0, 0)$, which no longer lies on the sphere.


So, with all that having been said to describe the difference between points and vectors, why do we sometimes assign vectors to points? Well, sometimes you can do this meaningfully--when space is "flat," so to speak. This is common in introductory physics, for instance. Very little about simple mechanics or electromagnetism puts you in a situation where you can't describe points with vectors. It is still, in my opinion, useful to distinguish between such so-called "position vectors" (which describe directions relative to the origin) and vectors that describe directions relative to some other fixed point.

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Here's an answer without using symbols.

The difference is precisely that between location and displacement.

  • Points are locations in space.
  • Vectors are displacements in space.

An analogy with time works well.

  • Times, (also called instants or datetimes) are locations in time.
  • Durations are displacements in time.

So, in time,

  • 4:00 p.m., noon, midnight, 12:20, 23:11, etc. are times
  • +3 hours, -2.5 hours, +17 seconds, etc., are durations

Notice how durations can be positive or negative; this gives them "direction" in addition to their pure scalar value. Now the best way to mentally distinguish times and durations is by the operations they support

  • Given a time, you can add a duration to get a new time (3:00 + 2 hours = 5:00)
  • You can subtract two times to get a duration (7:00 - 1:00 = 6 hours)
  • You can add two durations (3 hrs, 20 min + 6 hrs, 50 min = 10 hrs, 10 min)

But you cannot add two times (3:15 a.m. + noon = ??)

Let's carry the analogy over to now talk about space:

  • $(3,5)$, $(-2.25,7)$, $(0,-1)$, etc. are points
  • $\langle 4,-5 \rangle$ is a vector, meaning 4 units east then 5 south, assuming north is up (sorry residents of southern hemisphere)

Now we have exactly the same analogous operations in space as we did with time:

  • You can add a point and a vector: Starting at $(4,5)$ and going $\langle -1,3 \rangle$ takes you to the point $(3,8)$
  • You can subtract two points to get the displacement between them: $(10,10) - (3,1) = \langle 7,9 \rangle$, which is the displacement you would take from the first location to get to the second
  • You can add two displacements to get a compound displacement: $\langle 1,3 \rangle + \langle -5,8 \rangle = \langle -4,11 \rangle$. That is, going 1 step north and 3 east, THEN going 5 south and 8 east is the same thing and just going 4 south and 11 east.

But you cannot add two points.

In more concrete terms: Moscow + $\langle\text{200 km north, 7000 km west}\rangle$ is another location (point) somewhere on earth. But Moscow + Los Angeles makes no sense.

To summarize, a location is where (or when) you are, and a displacement is how to get from one location to another. They have both magnitude (how far to go) and a direction (which in time, a one-dimensional space, is simply positive or negative). In space, locations are points and displacements are vectors. In time, locations are (points in) time, a.k.a. instants and displacements are durations.

EDIT 1: In response to some of the comments, I should point out that 4:00 p.m. isn't at all a displacement, but "+4 hours" and "-7 hours" are. Sure you can get to 4:00 p.m. (an instant) by adding the displacement "+16 hours" to the instant midnight. You can also get to 4:00 p.m. by adding the diplacement "-3 hours" to 7:00 p.m. The source of the confusion between locations and displacements is that people mentally work in coordinate systems relative to some origin (whether $(0,0)$ or "midnight" or similar) and both of these concepts are represented as coordinates. I guess that was the point of the question.

EDIT 2: I added some test to make clear that durations actually have direction; I had written both -2.5 hours and +3 hours earlier, but some might have missed that the negative encapsulated a direction, and felt that a duration is "only a scalar" when in fact the adding of a $+$ or $-$ really does give it direction.

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4:00 p.m. is also a displacement = +16 hours from midnight. In fact any date is a displacement, xxxx year AD or BC - they all have an event, from which the calculation starts. –  Neolisk Jan 21 at 16:43
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@Neolisk 4:00pm is only a displacement when compared to another time, like midnight. It could also be said that the point (3,5) is a displacement when compared to another point like (0,0). –  EtherDragon Jan 21 at 16:59
    
@EtherDragon: That's exactly where I was going, which infers that point and vector are the same thing, data wise. –  Neolisk Jan 21 at 17:14
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@Neolisk, No problem, just a misunderstanding! Don't feel too bad, the two words are confused frequently enough by native speakers, and are sometimes used interchangeably in non-technical speech. They do have different definitions in English, though. :) –  Brian S Jan 21 at 19:34
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I think this is a really good explanation of why a point and a vector are not conceptually the same thing. It might be helpful to mention (if you agree) that in many cases (e.g. computer graphics) points are represented as vectors from an origin. EtherDragon touched on this in the second comment. –  Dr. Wily's Apprentice Jan 21 at 19:37

From my experience in computer graphics, the vector definition is redefined to accommodate 3D graphics needs (maybe 2D as well, I can't recall).

Usually an entity with more than one value is treated as a vector, so a point may contain (x, y) and have two values thus making it a NON scalar element.

However, this is entirely subjective and I could be totally wrong.

So I guess I'm trying to say that a point can be a vector via similar traits and used interchangeably context dependant.

For clarification, in C++ a class might be called PointXY with two variables (x,y) and it can also be called Vector2D with the same (x, y) values.

Hope that helps :)

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Since there is a trivial translation between vectors and the points they end at when based from the origin it`s a distinction without a difference in most contexts.

So if you want to be precise about it you can transform your points into the corresponding vectors, do your vector operations on those and transform them back to points again.

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There is a difference of definition in most sciences, but what I suspect you're asking about is a rather nice one-to-one correspondence between points in real space (say perhaps $\mathbb{R}^n$) and vectors between $(0, 0, 0)$ and those points in the space of $n$-dimensional vectors.

So, for every point $(a, b, c)$ in $\mathbb{R}^3$, there's a vector $(a, b, c)$ in the space of all 3-dimensional vectors.

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I think you are thinking about Geometry and Algebra, but in a broader mathematical sense, you can do ok by just talking about vectors and that's it. A Vector is a particular element in a vector space.

I a hilbert Space for example, a vector might be the Sinc function on the real numbers, and that is not a point in space, nor an arrow indicating direction or displacement or anything. It is just an abstract concept.

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It seems that the origin of the confusion comes from the representation of the point and a vector as a list of coordinates, which is usually indeed the same for both and when looking at the coordinates, say, a=(3,2,7) in 3D, there is no way to say is it a point or a vector.

It is easy to grasp the difference, with one more coordinate added to mark it: it is 0 for vectors, and 1 for a valid point: now there will be two valid cases: a=(3,2,7,0) means a vector, and a=(3,2,7,1) means a valid point.

Then it is clear, that any sum (+/-) of vectors will result in a valid vector; the difference of two points is a vector, vector added to another point results in another valid point. A special case, when weighted sum of points results in another valid point happens only if the extra coordinate is also 1 (sum of the weights=1).

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Even though this is math.stackexchange, I'll try to give you an intuitive, physics-y answer. Hopefully that will help you to tackle the formal mathematical side on your own.

I assume that you're thinking of a point as some position or place in space, which is fine to start with. Now imagine an arrow going from the origin of space to this point. That defines a vector. Every point in space will have a unique such vector, so there's no ambiguity if I said something like "the vector for the point P". But it works the other way, too, in that if I take any arrow that starts at the origin, it is associated with a unique point in space (the tip of the arrow). So there's no ambiguity if I say, "the point for the vector V". Since there's a one-to-one dictionary between vectors and points in space, we can go ahead and just define a point in terms of its vector, which is what I suspect is being done in the reference you're using.

That's the intuition, but as you can see from other comments and answers here, I'm leaving out technical detail and I refer you to them for the niceties.

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It would be nice if the downvoter could tell me what was wrong with this answer. –  josh314 Jan 21 at 20:20
    
I see nothing wrong with this answer... you even said it was physics-y so I upvoted it. –  Squirtle Jan 21 at 22:10
    
@Squirtle Thanks. I actually don't mind getting downvoted if I get some constructive criticism along with it. –  josh314 Jan 21 at 22:19
    
Problem here is that too many people downvote and don't justify.... and as a theoretical physicist in training, I see nothing wrong with this answer mathematically (or otherwise) –  Squirtle Jan 22 at 21:54
    
Well, my answer is not very precise or technical, so I could see how some people wouldn't like it. On the other hand, from the way the question was posed, I was intuiting that the OP is probably in the middle of undergrad, perhaps not even a math major, so I thought my answer would be more useful then some of the very detailed, technical responses that I saw. Anyway, I appreciate the support. –  josh314 Jan 23 at 6:53

An origin-based vector (the only kind that has only "direction and magnitude", as you write) can be represented by the point at its head. So there's a natural correspondence between points and vectors in this context, which is easiest to see if you throw in Cartesian coordinates so that both are expressible as a tuple of real numbers.

Just because one can represent the other does not mean that they are the same thing, however. You can talk about the distance between points, but sum and dot product are only meaningful for vectors. If you apply them to a "point", you're really just treating it as standing in for the corresponding vector. Conversely, points pop up in a lot of places where you cannot form a vector space. Think of $\mathbb Z \times \mathbb Z$: This is a (discrete) space, but not a vector space.

Consider also that a $2 \times 2$ matrix $\left [ a\, b \atop c\, d \right ]$ can be represented as the tuple $(a, b, c, d) \in \mathbb R^4$. So a point in $\mathbb R^4$ can represent a (regular) vector or a $2 \times 2$ matrix. What distinguishes them are the operations we define on them.

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On the Cartesian plane a vector represents a movement: left (by -x) or right (by x), and then down (by -y) or up (by y). In relation to a vector, a point is the grid location where that movement starts or ends. While still conceptually separated, a vector starting at the origin (0,0) and leading to a given point is numerically identical to that point.

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These are, no pun intended, categorically the same.

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A Vector (or it's coordinate representation) is invariant under translation of the origin; a Point (or it's coordinate representation) is not. If that isn't a clear distinction proving they are different I'll eat my hat(s) next December.

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