Point or Vector? What's the difference?

[Swapnil]

What's the difference between a point and a vector (both of which are situated in 3-Dimentions)? I mean, they are similar in a way that they can be represented as an ordered 3-tuple (x,y,z). But then they are different because vectors have a direction but points do not.

How would a mathematician differentiate between these two concepts?

 

[mathwonk]

a vector is an ordered pair of points. a point is one point.

 

[fourier jr]

a POINT is the point at (for example) (x,y,z). a vector is something with magnitude & direction. the VECTOR (x,y,z) is represented by an arrow starting at (0,0,0) & ending at (x,y,z). its magnitude is sqrt(x^2 + y^2 + z^2). it looks like an arrow but is actually a lot more. an arrow only has direction.

 

[CRGreathouse]

Since you can assume the vector starts at the origin, there's no more information in either and they have a natural bijection (the identity funvtion). Vectors are thought about in a different way, though. It doesn't make sense to add two points, but it does make sense to add two vectors. Similarly, a vector has a length and other properties that points are not though of as posessing.

 

[mathwonk]

greathouse is assuming the existence of an origin which you did not specify.

in a euclidean plane with no origin, a vector is an ordered pair of points. you cannot add two points but you can subtract two points, the resuklt being a vector.
'
you can also add a vector to a point.

 

[HallsofIvy]

You can add vectors, you can't add points. In particular, there exist a "0" vector but no "0" point. If you put a coordinate system on a set of points, then you can treat it as a vector space, but the connection between points and vectors depends on the coordinate system.

 

[mathwonk]

it is true you cannot add two points in an affine space, but there is no need for a coordinate system to add vectors to points.

to add a vector V to a point P, you start the vector at the point, and see where the end of the VEctor winds up. that point Q is V +P.

similarly to subtract P from Q you draw the Segment beginning at P and ending at Q, that is V = Q-P.

Of course you need an afine structure on your set of points but no origin and no coordinate system.

beginning from an affine space of points, including the ability to define parallelism and equality of lengths, you define a vector as an equivalence class of ordered pairs of points as usual.

or more elegantly, you deine the vector or tangent space of an affine space as the space of all length preserving bijections of the affine space with itself. then a vector tangent to A is an affine transformation
v:A-->A , and adding v to a point P of A means simply applying v as a transformation, v(P).

this is my favorite way to do it, and it was taught this way in my advanced calculus class in 1963.

in this form, Halls' observation about the zero vector as quite different from choosing any special point, is clear, i.e. the zero vector is the identity transformation on the affine space of points. it is the unique tranformation taking each point to itself.

 

[mathwonk]

by the way this makes it clear you do not need a coordinate system, or a notion of length, just the ability to compare lengths, to define derivatives.

i.e. since you subtract f(x)-f(a), you do nt need an origian, and since you divide the two segments [f(x)-f(a)]/(x-a), you do not need a unit length.

so you can deine derivatives by the usual formula for any function between copies of the affine line, or between affine spaces.

 

[CRGreathouse]

greathouse is assuming the existence of an origin which you did not specify.

This is true. I assumed the existence of a distinguised point from which vectors could be measured. It need not be the traditional Cartesian origin, of course; just about any point will do. Once you have it, you can define vectors with just as much information as points.

 

[mathwonk]

in the setup i have given above, once you pick any point O, then applying a vector transformation v to this point, gives the second point v(O) which may then be thought of as representing v with respect to the fixed point O.

conversely, with O fixed, for each choice of second point Q, there is a unique affine transformation (vector) v, such that v(O) = Q.


thus the affine space of popints and its intrinsically associated vector space of transformations, encapsulates all aspects of the more traditional but less natural versions of the connections between vectors and points.


this point of view was espoused by my calc teacher Lynn Loomis. I thought it very beautiful but I have essentially never seen it again in over 40 years.

 

[neurocomp2003]

mathwonk, outta curiosity what you posted about vectors, does that imply that the vector (0,0,0) need not exist in the vector space?
if I remember correctly the (0,0,0) must exist
if there exists a (0,0,0) vector doesn't that imply there exists a (0,0,0) point(or rather how can an euclidean space not have a 0,0,0)?

also aren't all points vectors added to the vector (0,0,0)?

BTW. What is the definition of a coordinate system?

 

[Swapnil]

in the setup i have given above, once you pick any point O, then applying a vector transformation v to this point, gives the second point v(O) which may then be thought of as representing v with respect to the fixed point O.
conversely, with O fixed, for each choice of second point Q, there is a unique affine transformation (vector) v, such that v(O) = Q.
thus the affine space of popints and its intrinsically associated vector space of transformations, encapsulates all aspects of the more traditional but less natural versions of the connections between vectors and points.

this point of view was espoused by my calc teacher Lynn Loomis. I thought it very beautiful but I have essentially never seen it again in over 40 years.

This is a pretty interesting point of view. My thanks to Lynn Loomis. BTW, what do you mean by "affine space"?


I am still a little confused. Things seems to be flying all over in my head. I know what's a point and what's a vector geometrically (the way most non-mathematicians think of points and vectors). But I don't know how to distinguish between them if I just stick to the concepts of objects and sets. It seems to me, from this discussion, that a vector only differs from point in the way that we can perform operations on vectors but we can't perform operations on ponits. Am I right?

 

[mathwonk]

if you took euclidean geometry, affine space is the euclidean plane, no origin no axes, just flat space, and the ability to detect parallelism and compare lengths.

 

[mathwonk]

neuro i could not understand your question. but maybe I am tired.

 

[Swapnil]

What about this?

I am still a little confused. Things seems to be flying all over in my head. I know what's a point and what's a vector geometrically (the way most non-mathematicians think of points and vectors). But I don't know how to distinguish between them if I just stick to the concepts of objects and sets. It seems to me, from this discussion, that a vector only differs from point in the way that we can perform operations on vectors but we can't perform operations on ponits. Am I right?

anybody?...

 

[mathwonk]

havent we been answering this all the time ? this makes me want to go back to my first answer, i.e. a point is a point and an ordered pair of points represents a vector,

two ordered pairs of points represent the same vector if the two oriented line segments are parallel and of the same length,.

 

[Michael Averasky]

Basically a point is one dot (with no dimensions). A vector however is in fact a quantity with both a magnitude and a direction. The key differences is the magnitude and direction. If you find the notation confusing, what we do using the Cambridge system of education is that we write vectors upright, sort of like matrix.

 

[mathwonk]

how can you confuse one point with two points?

 

[mathwonk]

here is a point: P. here is a vector: PQ.

the vector starts at P and ends at Q.

 

[Castilla]

If I have understood well:

A point: (2, 3).

A vector: ((h, k), (h+2, k+3)).

 

[radou]

If I have understood well:

A point: (2, 3).

A vector: ((h, k), (h+2, k+3)).

(2, 3) is an ordered pair. There exists a bijection between the Euclidean plane E^2
(which consists of points) and the Carthesian product of the reals R^2,
but that is a mapping between two sets of different objects. So, unless I'm missing
something big here, (2, 3) is not a point. An ordered pair is a vector, since
it is an element of a vector space. A point (in a strict geometrical sense) is not
an element of a vector space, and so it's not a vector.

 

[mathwonk]

the only reason you think (a,b) is a vector is because you are assuming the other point is (0,0).

so actually the vector is the pair (0,0), (a,b). get it?dont they tell you that vector can have its foot anywhere?

thats because <(0,0), (a,b)>, and <(c,d), (a+c,b+d)> represent equivalent vectors.

we do say that (a,b) are the "coordinates" of that common family of equivalent vectors.

think about it

 

[mathwonk]

more accurAELY, a pair of corrdinTE REPRESENTS MORE THAN A POINT. IT REPRESENTS A POINT IN RELATION TO SOME OTHER POINT,M HENCE 2 POINTS.SO P IS APOINT, (A,B) IS REPRESENTATOVE OF AN ORIGIN, AND A POINT LOCARTED RELATIVE TO THAT ORIGIN WRT A PAIR OF AXES.SO (A,B) STRICTLY SPEAKING IS NOT JUST A POINT. IT \IS A PAIR OF NUMBERS AND AS SUCH CAN RPRESENT MANY THIGNS.

 

[JasonRox]

more accurAELY, a pair of corrdinTE REPRESENTS MORE THAN A POINT. IT REPRESENTS A POINT IN RELATION TO SOME OTHER POINT,M HENCE 2 POINTS.


SO P IS APOINT, (A,B) IS REPRESENTATOVE OF AN ORIGIN, AND A POINT LOCARTED RELATIVE TO THAT ORIGIN WRT A PAIR OF AXES.


SO (A,B) STRICTLY SPEAKING IS NOT JUST A POINT. IT \IS A PAIR OF NUMBERS AND AS SUCH CAN RPRESENT MANY THIGNS.

In essence, we take for granted that when we write ordered pairs as vector, we are assuming that it's "foot" is (0,0) without even realizing it. (Well, I do, but most don't. )

 

[radou]

In essence, we take for granted that when we write ordered pairs as vector, we are assuming that it's "foot" is (0,0) without even realizing it. (Well, I do, but most don't. )

Of course, if we think of a vector as a radius vector. But I don't understand what that has to do with ordered pairs. Ordered pairs of, let's say real numbers, have an binary operation of addition and scalar multiplication defined, and they satisfy the general properties of a vector space. So, it makes them vectors. Of course, not vectors in the 'popular' meaning.

 

[mathwonk]

of course your concept satisfies the popular formal definition of "vector space" as defined in books, but i am trying to explain the actual concept of geometric vectors in euclidean space, which is the concept that has some content for physicists, and geometers too.for example, if one is trying to find the velocity vector of a moving point in the plane, what sense does it mean to say that velocity vector is (6,2)?the correct concept is to say the velocity vector at the time t, is a vector that begins at f(t) = p, and ends at f(t) + f'(t) = q, where f is the parametrization of the curve and f' is the derivative.i.e. it begins at the point p where the point is at time t, and ends at q, the point where the point WOULD be, after one unit of time, assuming t stopped accelerating at time t and continued in a riught line at uniform speed.obviously a geometric vector tangent toa curve, should have its foot ata poimt on the curbe where the tangency occurs. it is very artificial to have the foot of the vector at the origin, but that is forced by the formal belief that vectors are pairs of numbers.

think about it.

the point is that real vectors should be independent of the coordinate system or they have no physical meaning.

i admit this is confusing. i have even had to correct errors by full professors on this matter in responding to student complaints of their grading of several variaBLE CALCULUS EXAMS.

 

[mathwonk]

try to imagine a flat homogeneous plane, no coordinates, no origin. now what isa vector in that plane? (2,3) makes no sense.

 

[JasonRox]

Of course, if we think of a vector as a radius vector. But I don't understand what that has to do with ordered pairs. Ordered pairs of, let's say real numbers, have an binary operation of addition and scalar multiplication defined, and they satisfy the general properties of a vector space. So, it makes them vectors. Of course, not vectors in the 'popular' meaning.

Sure it does make them vectors. Why wouldn't it?

You have to remember that the addition and scalar multiplication of the vectors is the same as the regular ordered pairs. (Parallelogram)

The only thing is that when you make the ordered pairs a vector space, you are in fact using the origin as the "foot".

The ordered pairs also have a distance defined to them (using inner products), by using what? The origin itself.

But of course, you change shift the whole vector space (affine space) to the right so that (1,0) is the origin or "foot". In turn, this affects the operations of the ordered pairs and vectors in the exact same way.

 

[mathwonk]

here is a little thought problem: think of a uniform plane where one has no coordinates, but one can compare distances: i.e. here is no number asigned to a distnace but one can compare two lengths to se what scalar multiple one is of the other.

then define a derivative a follows; given two points p,q and a function f,

one can consider the ratio [f(q)-f(p)]/[q-p], which is a number.

then take the limit as q approaches p. this number is the derivative if f at p./im tired. ask yourself what is a god way to define the tangebnt vcector space to a uniform euclidean plane, withiout coordinates.

 

[Smacal1072]

I know this thread has been dormant a long time, but I think it should be revived, because vectors are so important to physical systems. They're always explained as a quantity having "magnitude and direction" in high schools and universities, which makes me sick. The students never really understand what they are, they just take them at face value. I didn't understand vectors completely even after a bunch of vector calculus.

I think throwing around more ideas on how to approach defining vectors is healthy.

Any takers?

 

[radou]

I think throwing around more ideas on how to approach defining vectors is healthy.

Any takers?

There are no ideas. The only "healthy" definition of a vector is that it's an *element of a vector space*. That's pretty much all of it.

 

[jostpuur]

I have never seen vectors being defined as ordered pair of points.

 

[radou]

I have never seen vectors being defined as ordered pair of points.

And you will never see that, either.

 

[HallsofIvy]

mathwonk, outta curiosity what you posted about vectors, does that imply that the vector (0,0,0) need not exist in the vector space?
if I remember correctly the (0,0,0) must exist
if there exists a (0,0,0) vector doesn't that imply there exists a (0,0,0) point(or rather how can an euclidean space not have a 0,0,0)?

also aren't all points vectors added to the vector (0,0,0)?

BTW. What is the definition of a coordinate system?

mathwonk didn't say anything like that! Any vector space must contain a 0 vector. If you have just points, then you can construct the vector between any ordered pair of points. The vector you construct from (P, P) where P is a single point is the zero vector.
I made the the point that in order to put a vector space on a space of points (an "affine" space) you need a coordinate system. mathwonk objected to that, saying you can construct vectors between any two points. That's true. What I meant to say was that to ASSIGN one vector to each point you need a coordinate system. That's wrong to! As mathwonk said, you just need a single "given" point, O, and you can then assign to every point P, the vector from O to P. In particular, the vector from O to itself is the 0 vector. Of course, you could then choose an orthonormal basis for the vector space and THEN you have a coordinate system!

I am still a little confused. Things seems to be flying all over in my head. I know what's a point and what's a vector geometrically (the way most non-mathematicians think of points and vectors). But I don't know how to distinguish between them if I just stick to the concepts of objects and sets. It seems to me, from this discussion, that a vector only differs from point in the way that we can perform operations on vectors but we can't perform operations on points. Am I right?

Not entirely. because a vector can be thought of as "from one point to another", vectors have more "structure" than points.

 

[romistrub]

I found this thread while googling "vector vs point".

In the context of vector spaces:

If a vector space \V is defined over a field of scalars |F (which *might not* be the real or complex numbers we're used to dealing with http://en.wikipedia.org/wiki/Field_(mathematics)#Examples"), then a "point" is an ordered tuple *of scalars* representing the expansion of the linear vector in a given basis (i.e. it's coordinates in a basis).

Any vector |v> in the vector space can be represented as a linear combination of the basis vectors. In general, in a two-dimensional vector space (and in the most abstract form you can probably think of):

\vec{v}=(\alpha_1 \hat{X} \vec{e}_1) \hat{O} (\alpha_2 \hat{X} \vec{e}_2)

Note that the scalars (alphas), operators (X and O) and vectors can be defined in *any way at all*. So long as they obey the axioms of a vector space, the entirety of vector analysis can be applied on them. I used two dimensions just to simplify the notation, but the same can be said for any number of dimensions.

For example, given the two-dimensional vector space R² (defined over the field of real numbers, no less), and a basis {|e>} which contains the vectors:

|e1> = (2,0)
|e2> = (0,3),

and a very important vector (not for any good reason, just because I had to pick one):

|v> = (8, 15),

then |v> is expanded in this basis as |v> = 4*|e1> + 5*|e2> and its coordinates in this basis are represented by the n-tuple (4, 5). In our particular basis, given our particular vector, the idea of expansion is natural. This is not always the case.

What is "the origin" in this vector space? It's not the 0-vector: the 0-vector |0> is the vector that, when added to any other vector |v>, results in |v>, which is not related to the idea of an origin. This property would be true even if the notion of "origin" didn't exist.

An origin is represented as the coordinates (0,0,0), where 0 is the scalar additive identity. In a vector space, these coordinates result in the |0> vector, regardless of the basis used, since the multiplication of any vector by the 0-scalar is the |0> vector. Hence, the idea of "shifting the origin" cannot be thought of as a change in the coordinates (0,0,0) to some other coordinates, nor a change in the property that these coordinates result in the |0> vector.

A shift in the origin is no shift in the origin at all. It is represented by a http://en.wikipedia.org/wiki/Translation_(mathematics)" on all vectors in the vector space, so that the coordinates of each vector are shifted by some amount relative to the origin.

What does this have to do with "points"? Well, a point is an ugly notion in a vector space, but I suppose it can be thought of as a set of coordinates in a basis. What is the relation between the points (1,1) and (2,2), then? Without a notion of basis, it means nothing. In a basis, each of these points represents exactly one vector and hence can be represented as such.

Just as the notion of points in vector space is ugly, so is the notion of vectors in Euclidian space. And just as the notion of a vector |v> in a vector space is independent of any basis (and hence independent of any coordinates or "origin"), so is the notion of a point independent of that of basis (and coordinates and origin) in Euclidian space.

Adding and subtracting is always defined in a vector space, since the vector addition operation is a requirement for the vector space, as is the |0> vector. Moreover, when a basis is defined, it is possible to add vectors by adding their coordinates in this basis. However, no such constraint exists for the existence of a (0) point (the origin) in Euclidian space. So how do you add points in E-space? It's not always possible (to see more, check out the idea of an http://en.wikipedia.org/wiki/Affine_space" ).

So that's the difference. Euclidian space is an n-tuple space (or a coordinate space), whereas a vector space, while it *can* be an n-tuple space (as our example above), doesn't have to be, and coordinates in a vector space are not the fundamental component of a vector space as they are of the Euclidian Space. Why do we always talk about vectors in Euclidian space? Note, first, that vectors in Euclidian space are universally considered as "arrows" or some other representation of displacement. This is the first clue, and it is reinforced by assigning a "head" and "tail" to the vectors. Any time you consider the relationship between two points as a vector in Euclidian space (and any relationship between vectors in Euclidian space), you construct a transient vector space in which the axioms are satisfied by your arbitrary assignment of a |0> vector (whichever point you picked as your reference point).

The confusion compounds because the notions of inner product, orthogonality, and all that other fancy stuff exists in Euclidian space. This can be considered as either intrinsic to Euclidian space, or just the result of the transient vector space. In the former case, the Euclidian space has absolutely *no* relation to any vector space at all, and all its properties are built from the ground, up, including the difference between the notion of vectors and points.

It gets even messier when dealing with vector-valued functions defined over Euclidian space. That is, the typical "vector fields". Are the vectors defined in the Euclidian space? Not really. They belong to a vector space, and those neat arrow-diagrams represent a "graph" of the function (consisting ((x,y), V(x,y)) where V(x,y) is a vector. Transfering the results from the vector space to Euclidian space is just the reverse of transferring points from coordinate space to vector space. Note that the function V(x,y) can also be considered to produce a point relative to (x,y), in which case the function maps the Euclidian space into itself, and the set of points and their associated outputs from V can be interpreted as vectors (or not) as you please. A counter-example is simply a function that accepts coordinates and returns other coordinates which are not intended to be considered relative to the input coordinates.

 

[romistrub]

I should add that it's even more apparent how we use vectors in geometric/Euclidian contexts when you think about how basis vectors are defined in http://en.wikipedia.org/wiki/Curvilinear_coordinates" : the basis vectors are actually a function of the coordinates by which a space is described. Because of this, they are called a "local basis" instead of a "global basis" (although a global basis is really just a special case of a local basis that is invariant with coordinates). This is evidence that, when talking about a vector in geometric space, we create a transient vector space whose basis vectors depend on the point we are interested in, and we describe the vector in this basis. Hence, comparing vectors requires an understanding of the bases on which the vectors are defined.

For example, in Euclidian space, the vector

|v1> = 4*|i> + 2*|j> + 3*|k>

is generally unambiguous since, even though no vector tail is specified, it defines a unique direction in Euclidian space. Whereas in spherical coordinates (my favourite), with points given by (r, th, ph) and basis vectors |r>, |th> and |ph>, the vector

|v1> = 4*|r(r, th, ph)> + 2*|th(r, th, ph)> + 3*|ph(r, th, ph)>

is not only point-ambiguous, but direction-ambiguous, since the basis vectors have not been specified because the origin/tail/start of |v1> has not been specified. This "vector" is actually a vector field defined for all (r, th, ph), whose components are always (4, 2, 3) in the direction of |r> |th> and |ph>, whatever that direction may be at each (r, th, ph).

Hence, for the most part, we (I...) rarely talk about vectors as isolated entities in curvilinear coordinates, but instead refer to a vector-valued function of coordinates: since the input of the function is the coordinates (r, th, ph), the basis vectors are uniquely determined, as is the direction of the vector. If we want to specify a single vector, we specify the vector function at a specific coordinate and hence simultaneously determine the tail and direction of the vector.

If a vector field V defined by a vector function V(r, th, ph) is dependent on the coordinates (r, th, ph), it is possible to ensure that these vectors are invariant across a change of coordinates by translating V(r, th, ph) to V(x, y, z) (for example) accordingly. Hence, the crucial invariance property of vectors can be preserved.

What's interesting is that, without coordinates, the notion of a point in Euclidian space is inseparable from a vector in a vector space: as soon as you define a point, you've analogously defined a |0> vector for a vector space by which you are free to define subsequent vectors (or related points). The "affine structure" of Euclidian space is important in that you can specify certain linear combinations without an origin, but all that really says is that certain linear combinations are equivalent to... specifying an origin (at which point you create a transient vector space and may apply the rules of vector spaces to create a new "point").

The only time a difference between points and vectors arises is when you are modeling a system and you decide to call one thing a point and another a vector. Your point is still a "displacement vector" of sorts, but it's easier to think of it as separate from vectors that you are defining *at* points. As such can also think of Euclidian space as a coordinate vector space P containing points, defined with a map (not necessarily bilinear) to another vector space V containing vectors.