I thought I'd bring this question to math.SE, as it could spark some interesting discussion.
When I first learned vectors - a long time ago and in high school - the textbook and teachers would always introduce them as a "quantity with both a magnitude and direction"
This definition always seemed to irk me. It seems to favor the "polar" definition of a vector and then teach "but it has Cartesian components [x_0, x_1 ... x_n] like so"
I felt that I reached a personal breakthrough when I realized that "A vector is a higher dimensional generalization of a 'number' or 'value' or 'quantity' "
This has been how I've always thought of them.
For example, the number five can be thought of as a one dimensional vector - <5>.
< . . . . . 0---------> . . .> 0 5
Just like the vector <5,5> is a two dimensional analog of this notion. It just so happens that the circumstances change in subtle ways - (the ability to have a 'distance' or 'absolute value' different from either of the components and the notion of a direction when graphed).
Also, it seems that every teacher is very very careful to distinguish a vector from a point. However, this seems trivial in concept, as we wouldn't distinguish "the position '5' on a number line" from "the value 5" or "distance from zero to 5"
Why is the 'magnitude and direction' definition favored?