Unizor - Vectors - Introduction
Channel: Zor Shekhtman on UNIZOR Education
Date: 10/12/2013

Mathematics is about abstraction. Thus, natural numbers is a mathematical abstraction of a concept of quantity. Introduction of rational and irrational numbers allowed to accommodate such concepts as length, distance, weight. Further on, negative numbers could be used as an abstraction of such concepts as forward and backward or increase and decrease.

Let's expand this process of abstraction to concepts related to movement. Consider a difference in meaning between words distance and displacement. Traditionally, distance is a measure of how far one object is from another, while displacement characterizes both how far and to which direction one object is from another.
Similarly, consider words speed and velocity. The former measures how fast an object moves while the latter takes into consideration both how fast and to which direction it moves.

Mathematics generalizes these and many other similar cases into concepts of scalars and vectors. The term scalar is used to quantify the magnitude of some physical process (like distance or speed) while the term vector combines both the magnitude and some directional property of the process. It is especially obvious when we talk about movement where the direction of this movement is a subject of discussion inasmuch as the speed or distance.

Going back to abstraction, mathematics not only introduces the abstract concepts but also implements them in some way to be able to manipulate with them. Thus, numbers might be implemented in some numerical system, usually decimal for humans or binary for computers. We know how to add and multiply them. We can also implement numbers as points on a coordinate line or, in case of small positive integer numbers, as a set of chips or other objects. So, an abstraction needs some implementation.

What's the implementation of vectors? Firstly, the magnitude is a part of a concept of a vector and we can represent it as a number using one of the known methods. Secondly, since the direction is also a part of a vector concept, we have to define its implementation.

The obvious difficulty in this implementation is to understand that a concept of direction is related to a concept of space, where the movement (the primary physical process that involves direction) takes place. We can move along the line or curve - a one-dimensional space where position is determined by one number. We can move on the plane or surface - two-dimensional space that needs two numbers to define position. We can fly to the Moon in a rocket - three-dimensional space with three coordinates required to determine the position. We can take time into account as the forth dimension like in Einstein's Theory of Relativity. All the above are just different examples of space as a mathematical abstraction. The concept of direction is applicable to each and every one of them and, therefore, when talking about vectors, we always have to keep in mind the space where these vectors are pointing the direction.

To summarize, talking about vectors, we always have to keep in mind the space where our vectors are located, their magnitude and direction in space they point to.