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The definition of a vector is usually something like "a quantity that has both a magnitude and a direction". But, in the context of, say, economics rather than physics, does this definition make sense? Let's say we are working in 4 dimensional space? Lastly, how does this definition make sense in matrix algebra when referring to a column vector, say, taken from a list of the number of vehicles sold by brand?

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    $\begingroup$ Have you seen the definition of "a vector space"? That's usually what mathematicians are talking about when they say "vector" (and in fact is also what physicists are talking about - "a quantity that has both a magnitude and direction" is really only useful insofar as distinguishing vectors from scalars, but not terribly useful for actually defining what a vector is) $\endgroup$ Aug 13 '15 at 2:50
  • $\begingroup$ Related. $\endgroup$ Aug 13 '15 at 3:25
  • $\begingroup$ Incidentally, if the high-school physics characterisation of a vector (in $\mathbb R^2$ & $\mathbb R^3$) as “a quantity with a magnitude and direction in space” is meant to contrast vectors with scalars, then the corresponding characterisation of a scalar really ought to be “a quantity with a signed value and no direction in space”. $\endgroup$
    – ryang
    Apr 9 '21 at 6:36
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The general definition of vectors do not require that a vector has a defined magnitude. It just so happens that in many of the common vector spaces that are useful for physics and sciences that there is a natural choice for a norm, making these spaces normed spaces (which are a special kind of vector space).

In general a vector is merely an element of a vector space.

A vector space is a set $\mathbb{X}$ such that the following properties hold:

  • For every $x,y\in\mathbb{X}$, you have also that $x+y\in\mathbb{X}$ (i.e. it is closed under addition)
  • For every $x\in\mathbb{X}$ you have $\alpha x\in \mathbb{X}$ where $\alpha$ is an element of an underlying field (commonly $\mathbb{R}$ or $\mathbb{C}$) (i.e. it is closed under scalar multiplication)
  • There is a "zero element", $0\in\mathbb{X}$, such that for each $x\in\mathbb{X}$ you have $0+x=x=x+0$.

Also required are things about the commutativity and associativity of addition, and distribution of scalars over vectors as well as distribution of vectors over scalars.

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    $\begingroup$ there are a few more requirements, but mostly intuitive en.wikipedia.org/wiki/Vector_space#Introduction_and_definition $\endgroup$
    – Elliot G
    Aug 13 '15 at 2:58
  • $\begingroup$ @ElliotG Indeed. It is rare as a student to be in such a situation that those additional requirements aren't automatically satisfied, so much so that I often forget to mention them. $\endgroup$
    – JMoravitz
    Aug 13 '15 at 3:00
  • $\begingroup$ didnt doubt you knew; just pointing out for OP $\endgroup$
    – Elliot G
    Aug 13 '15 at 3:01
  • $\begingroup$ I would like to accept this answer, but it would be pretentious to do so, as I don't understand 100% of it. I'm learning algebra, and now generally worried that I'm learning the methods just fine, but the intuition is not there. Math seems to be taught a bit like paint by numbers. Perhaps the intuition is just too hard for the average student, so it gets distilled to a bunch of rules....? I will study your answer deeply and see if it clicks. Thank you. $\endgroup$
    – Hexatonic
    Aug 14 '15 at 3:13
  • $\begingroup$ @P.Jakobsen It is like asking "what is a highway car." It is a car that can travel on a highway. While it so happens that most highways have speed limit signs and exit ramps and such, we don't require that those such things exist for the road to be called a highway. In the same way, a vector is an element of a vector space, and while most vector spaces have a concept of magnitude and direction, not all do. $\endgroup$
    – JMoravitz
    Aug 14 '15 at 3:20
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That is not a very general definition of a vector, no. But it is hardly "wrong" for that reason.

Direction and magnitude are intimately tied to spatial intuition.

The magnitude referred to is usually a "length" measured by a real number, but there are vector spaces where that idea makes no sense.

The same goes for "direction" too. You can, at best, only get a very primitive notion of direction, as discussed here

The best definition of a vector is probably "element of a vector space," with all the axiomatic behaviors implied. That does not mean it must be particularly useful for developing intuition about what a vector "is."

The direction-magnitude is a particularly apt way of describing vectors in 2 and 3 dimensional geometry, so it has its uses. After convincing yourself about higher dimensional real spaces, and the spaces over other fields, you will probably find yourself abandoning direction and magnitude in favor of the most general definition.

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