>> Interesting. It appears that we are ran into a mathematical >> cultural difference. Were I come from vectors *are* defined as >> having four properties that I enumerated. After some research I >> found that English sources (Wikipedia) indeed give the definition >> you supplied. >> > Indeed, interesting. > After some additional research I found that the most concise definition given here is 'A vector is an ordered pair of points'. From that follow the 4 properties: if we describe a vector by 2 points A(x_1, y_1) and B(x_2, y_2) we can convert this representation into the form [x_2 - x_1, y_2 - y_1] but some information is then lost in the process - conversion in the opposite direction is not possible without knowing the starting point. The article on the English Wikipedia (again, http://en.wikipedia.org/wiki/Vector_%28spatial%29) says: "*vector*, is a concept characterized by a magnitude and a direction. A vector *can be thought of as* (emphasis mine - G.) an arrow in Euclidean space, drawn from an *initial point* /A/ pointing to a *terminal point* /B/." Apparently what you know as the definition is a property where I come from, and vice versa.
On a side note, I discovered more such differences (my studies are exclusively in English and we use mostly English textbooks even though I'm not in an English-speaking country). According to the article in the English Wikipedia a set is: "Definition (...)By a /set/ we understand any collection /S/ of definite, distinct objects /s/ of our perception or of our thought (which will be called the /elements/ of /S/) into a whole." In 'Basic Concepts of Mathematics' (available for free at http://www.trillia.com/zakon1.html), the author (who was, BTW, a Russian that settled in Canada) states: "A set is often described as a collection (“aggregate”, “class”, “totality”, “family”) of objects of any specified kind. However, such descriptions are no definitions, as they merely replace the term “set” by other undefined terms. Thus the term “set” must be accepted as a primitive notion, without definition." To its credit, the Wikipedia article states that it describes the 'naive theory of sets'. Not a word about primary notions under http://en.wikipedia.org/wiki/Axiomatic_set_theory either, though. Similarly, I was once given a definition of a number as 'the common feature of sets with the same size' (or something to that effect), one that I couldn't find in any English source. Another, trivial, example can be the Abel theorem which is known as Buniakovsky (Buniakowski) theorem in Eastern Europe. There are also some inconsistencies in mathematical notation that I couldn't settle with my university lecturers but I don't want to deviate from the subject any further. I suppose they're the kind of doubts you get after reading too much Dijkstra ;) > I didn't say that you must not know the point of application, but I > said that it was not a property of the vector itself. It is true, > however, that in physical calculations you should not "mix" many > types of vectors (like force) that are, in the experiment, applied > to different points of application. > > Here it would be an intrinsic property of a vector ;). I think we've nailed the crux of the argument now. >> Again, I think we were given different definitions. Mine states >> that direction is 'the line on which the vector lies', sense is >> the 'arrow' and magnitude is the 'length' (thus non-negative). The >> definition is separate from mathematical description (which can be >> '[1 1] applied at (0, 0)' or 'sqrt(2) at 45 deg applied at (0, 0)' >> or any other that is unambiguous). >> > > Oh, I thought we were talking about quite mathematical vectors? > But we were, I think? >> No. In one-dimensional 'space' direction is a ± quantity (a >> 'sense'). In 2-d it can be given as an angle. >> > > Indeed, you're right. So, those vectors have different properties > depending on the used coordinate system? I myself prefer the > concise definition ... > I think it's similar to points in space: depending on the number of dimensions you need a corresponding number of coordinates to describe a point. Whether these are Cartesian or polar coordinates (complex plane also springs to mind) has no bearing on their number. Regards, Greg -- http://mail.python.org/mailman/listinfo/python-list
