Point or Vector? What's the difference?

[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.