Point or Vector? What's the difference?

[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.