Definition of a Vector

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