Difference between Euclidean space and vector space? I often hear them used interchangeably ... they are very complicated to make any use of.
Wikipedia words:
Euclidean space:

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle.

Vector space:

A vector space is a mathematical structure formed by a collection of elements called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context.

They are not related at all. A vector space is a structure composed of vectors and has no magnitude or dimension, whereas Euclidean space can be of any dimension and is based on coordinates.
I hear 3-D programming uses vectors, so Euclidean geometry should be useless, no?
Basically, aren't they unrelated?

Vector space is just 'space' of objects following certain rules: a+b=b+a, (a+b)+c=a+(b+c), k(a+b) = ka + kb, and so on. k should be from another 'space' with certain rules, such as ℝ
Normed vector space is a vector space with "norm" function defined, ||a||, which corresponds to our intuitive "size of a vector" and follows simple rules: ||a+b||<=||a||+||b||, etc.
Euclidean space is the Normed Vector Space with coordinates and with Euclidean Norm defined as square root of sum of squares of coordinates; it corresponds to our intuitive 3D space, Pythagorean theorem, angle between lines definition, 5th axiom of Euclid (and Minkovski provided example of different space where 5th axiom is not true), and so on; it corresponds to geometry which we learn at school and touch in real life.
There are many other examples of normed vector spaces: Minkovski (used in Einstein theory of relativity), Banach, Hilbert, etc.
