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?
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.
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.
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.
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.
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.
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.
This is true. I assumed the existance 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.
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.
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?
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?
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.
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,.
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.