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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?

  • 1
    Likely because it's easy to remember and it appeals to the geometric nature in which vectors and vector spaces are typically first introduced. This is one of those things where you've realized something great, but if you continue to learn math, you'll find that thinking of vectors as high-dimensional numbers doesn't work. To be fair, though, having 'magnitude and direction' isn't really good either.2012-08-22
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    For what it's worth, I _do_ distinguish the position $5$ on a number line, the value $5$, and the distance from $0$ to $5$. (The first is a point in a topological space (http://en.wikipedia.org/wiki/Topological_space), the second is an element of a ring (http://en.wikipedia.org/wiki/Ring), and the third is an element of a totally ordered set (http://en.wikipedia.org/wiki/Total_order).)2012-08-22
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    The reason for treating vectors as arrows is not technical but rather _pedagogical_. Many vectors that arise in applications are entities that have both magnitude and direction (position relative to a fixed origin, velocity, acceleration, etc.). As some of the answers have already indicated, vectors as mathematical objects are more abstract than this, but it's good to learn about them first in the concrete setting of vectors in $\mathbb{R}^n$, where thinking of them as arrows is very convenient (for example, it gives sense to the addition rule for vectors in $\mathbb{R}^n$).2012-08-22
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    If you are a physicist or an engineer vectors often represent things which are really measurable. I am not sure what is wrong, in this context, with "magnitude and direction" - writing physical equations in vector form expresses their invariance under certain transformations of basis (rotations, for example).2012-08-22

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