I understand that a vector has direction and magnitude whereas a point doesn't.
However, in the course notes that I am using, it is stated 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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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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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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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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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.
An analogy with time works well.
So, in time,
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
But you cannot add two times (3:15 a.m. + noon = ???)
Let's carry the analogy over to now talk about space:
Now we have exactly the same analogous operations in space as we did with time:
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. Displacements 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. is NOT 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 text 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.
EDIT 3: A summary in table form:
| Concept | SPACE | TIME |
|---|---|---|
| LOCATION | POINT | TIME |
| DISPLACEMENT | VECTOR | DURATION |
| Loc - Loc = Disp | Pt - Pt = Vec | Time - Time = Dur |
| $(3,5)-(10,2) = \langle -7,3 \rangle$ | 7:30 - 1:15 = 6hr15m | |
| Loc + Disp = Loc | Pt + Vec = Pt | Time + Dur = Time |
| $(10,2)+ \langle -7,3 \rangle = (3,5)$ | 3:15 + 2hr = 5:15 | |
| Disp + Disp = Disp | Vec + Vec = Vec | Dur + Dur = Dur |
| $\langle 8, -5 \rangle + \langle -7, 3 \rangle = \langle 1, -2 \rangle$ | 3hr + 5hr = 8hr |
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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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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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A Vector (or its coordinate representation) is invariant under translation of the origin; a Point (or its 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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Point is generic in nature, so one has scope to apply his/her subject expert and situation requirement to introduce various algebraic, geometric, or physical structure and many more.
While a vector is specific in nature, so one has to know the underline algebraic structure (the vector space), where characteristics of the vector space such as inner product, norm, basis, dimension, other important results play important role in application of the vector.
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: $\vec{ab}\times \vec{ac}$.