The following link to my book called
Understanding Vectors, provides some insights into a better understanding of vectors:
http://sites.google.com/site/newmathorg/Home/knol-and-blog-attachments/vector.pdf
Introduction:
It is not uncommon for educators and students alike to use vector terminology incorrectly.
Understanding Vectors contains sufficient information for a good basic understanding of vectors with real world applications.
What is a vector really? A ubiquitous definition is: A vector is a quantity that has magnitude and direction. At first inspection this definition appears to be quite harmless, but with further investigation one realizes it is confusing and ill-defined.
While it is true a given vector has a magnitude and a direction, the following is often misunderstood or overlooked:
-A vector's magnitude and direction are dependent on its components.
-A vector's components are determined primarily by the definition of the components
it carries and are quantified by its magnitude.
-All the components of a vector must be of the same type, that is; they are defined
by a magnitude and appropriate units of measure.
-A vector's direction is determined by its components. Components can be referred to
as direction numbers in one sense.
-The number of components in a vector is determined by the attributes of the
coordinate system in which it is applied.
-A vector has no position. As such it cannot be a point or a location.
Consider a one-dimensional vector whose component is defined as velocity. Its magnitude and direction in this case are the value of its component. As an example, if a car travels 50 mph east in a straight line, its magnitude is 50 mph and its direction is +50 or east.
Quantity means an indefinite amount or number. One who is knowledgeable in mathematics or science will agree this can only be true of a vector in one dimension, that is, a real number. To suggest that vectors in higher dimensions are a quantity, is not only absurd, but moronic. Two or three-dimensional vectors consist of more than one quantity or component.
Terminology is a source of great confusion. The following statement can often be seen in textbooks or references:
A boat moves with a velocity p. The wind is blowing with a velocity q. Therefore, the resultant velocity vector is p+q.
The word velocity has exactly the same meaning as speed. Webster's defines velocity as quickness of motion or the rate of change of position along a straight line. Using the expressions velocity and velocity vector in a sentence that suggests these are the same, is not only incorrect but also useless.
How is velocity p different from velocity vector p? What is meant by velocity and velocity vector in the previous statement? If one is to make sense of these expressions given both are valid, then the above statement should be written as follows:
A boat moves with a velocity |p|. The wind is blowing with a velocity |q|. Therefore, the resultant velocity vector is p+q.
If velocity and velocity vector are to have the same meaning, then the statement would be written best as follows:
A boat's motion is described by velocity magnitude |p|. The motion of the wind blowing is described by velocity magnitude |q|. Therefore, the resultant velocity vector is p+q.
One can see that ambiguity abounds in the way academics use and understand the word vector. The intent of this Knol is to clear up some of the confusion and make it easier for a learner to understand vectors; to wit, that quantities or magnitudes and direction are described in terms of their components.
John Gabriel's definition of vector:
A vector is a set whose elements are components such that a direction and vector magnitude can be determined from these elements.For example, (3 mph; 4mph) is a velocity vector in a plane whose components are velocity magnitudes. The resultant velocity has a magnitude of 5 mph and a direction defined by arctan(4/3). Velocity is the rate of change in distance with respect to time (the derivative of a vector-valued distance function). How is this different to speed? One may consider speed and velocity identical with the exception that direction is generally ignored with regard to speed.
Some incorrect vector definitions:
"A vector is a quantity that has both magnitude and direction. It is typically represented symbolically by an arrow in the proper direction, whose length is proportional to the magnitude of the vector. Although a vector has magnitude and direction, it does not have position. A vector is not altered if it is displaced parallel to itself as long as its length is not changed." - Encyclopaedia Brittanica
"A vector is a variable quantity that can be resolved into components." - Princeton (http://wordnetweb.princeton.edu/perl/webwn?s=vector)
"Mathematically, a vector is a quantity, defined by both magnitude and direction.
For example, a vector could be illustrated by an 1 inch arrow pointing at a 30 degree angle." - TechTerms.com
"A member of a vector space." - FOLDOC (Free Online Dictionary)
"A quantity that has magnitude and direction and that is commonly represented by a directed line segment whose length represents the magnitude and whose orientation in space represents the direction." - Merriam Webster
"Euclidean vector, a geometric entity endowed with both length and direction; an element of a Euclidean vector space. In physics, euclidean vectors are used to represent physical quantities which have both magnitude and direction, such as force, in contrast to scalar quantities, which have no direction." - Wikipedia
Wikipedia takes the prize for most absurd definition of vector. Given that all geometric concept of point (location), which is decidedly not an entity, it follows that a vector cannot be an entity. Why is a point not an entity? A point is not an entity because a location has no independent, separate or self-contained existence. Paradoxically, it is the requirement of a separate entity or a different location that validates the existence of a given location.
For more thought-provoking ideas, one can read chapter 1 of my calculus book at:
http://sites.google.com/site/newaveragecalc/Home/ch1.pdf
In higher dimensions, it is still possible to think of vector magnitude as defined by the distance formula. Although direction takes on a different meaning, one can assume the components are themselves direction numbers. As for how direction is defined, the answer to this question remains open-ended. Direction is defined differently in all dimensions. For example, in one dimension, there is one of two directions. In two dimensions, there are 2π radians. In three dimensions, the model changes completely to a radial vector concept; that is longitude and latitude.
"An educator understands only what he can explain and can explain only what he understands." - John Gabriel
It is my opinion the remainder calculated in both cases is incorrect in the Britannica. 
Cotes began his research using Lagrange's interpolation polynomial which for all intents and purposes is the same in functionality and results as Newton's divided differences polynomial.
