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### Definition of a Vector

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Date: 02/28/2002 at 20:15:42
From: Jim Graziose
Subject: Definition of a Vector

I recently came across the statement "a vector has magnitude and
direction." I believe that this statment is incorrect and I would
like to acquire a "proper" definition of a vector and how concepts
of "magnitude" and "direction" arise in the discussion of vectors.

From my college undergraduate days, I recall that a vector is an
element of a vector space. So far I've done searches on the Internet
as well as looking at linear algebra and physics texts with no luck.

Thank you again for your help.
```

```
Date: 03/02/2002 at 13:05:45
From: Doctor Roy
Subject: Re: Definition of a Vector

Hello,

Thanks for writing to Dr. Math.

It's a matter of interpretation. Physical vectors (those used in more
practical applications) can be described fairly well by quantities
with a magnitude and a direction. These are a specialized case of the
more general mathematical concepts of vectors and vector spaces. But
if you want the more general definition, here is one:

Let's say we have a set X. Then, X is a vector space and all elements
of X are called vectors if the following properties hold (over a field
of scalars F).

For all x and y in X, x + y = y + x
(i.e. commutativity holds).

For all x, y, and z in X, (x + y) + z = x + (y + z)
(associativity).

There exists an element e in X such that x + e = e + x = x
for all x in X.

For each x in X, there exists a y in X, such that
x + y = y + x = e.

For each x in X, id*x = x   (id in F)
Note: multiplication indicates the element is taken from your
field of scalars. Fields are a more complicated matter. It
suffices to say they are likely to be integers or reals or maybe
complex numbers.

For each pair a,b in F and each element x in X
(ab)x = a(bx)

For each a in F and each pair x, y in X
a(x + y) = ax + ay

For each pair a,b in F and each x in X
(a + b)x = ax + bx

We call (x + y) the sum of x and y and ax the product of a and x.

This is a fairly rigorous and dry definition, but it is exact.

The concept of magnitude and direction involves other concepts for a
rigorous definition. But let's simplify. Let's say we want to know how
much a vector corresponds to another. If two vectors are parallel, we
would expect them to correspond highly. Conversely, if two vectors are
orthogonal, we would expect them to have no correlation. This brings
about the concept of an inner product, which satisfies properties
common to a metric (or distance).

This, of course, begs the question, "How much does a vector correspond
to itself?" The result is the magnitude, or norm, of a vector. As for
direction, we may want to know that normalized to the same length, how
close are two vectors to parallel (or perpendicular). So, independent
of magnitude, how "close" are two vectors to coinciding? This leads to
direction. Of course, direction is defined for vectors having more
than two dimensions, so it may mean little in a purely physical
interpretation.

I hope this helps.

- Doctor Roy, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 03/02/2002 at 14:35:52
From: Jim Graziose
Subject: Definition of a Vector

Dear Dr. Roy.

I really appreciate the information you furnished me about the
definition of a vector.

Best wishes,
Jim Graziose
```
Associated Topics:
High School Linear Algebra
High School Physics/Chemistry

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