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
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