Definition of Vectors

 

Introduction

In life there are objects that can be fully described by only one number. For example, if we want to know how much money we have in our pocket (liquidity), only one number is sufficient to inform us. 

For other type of objects this is not enough. For example, if we want to describe the force that is applied to an object, we have to define the magnitude, as well as the direction of the force that it is applied. The packet containing magnitude and direction is called vector.

A vector is usually noted by one or more characters with an arrow over them. So if we want to note the vector of force F, we would write:

Vectors can be described geometrically. They are designed as arrows. The magnitude is represented by the length of the arrow and the direction by the angle between the arrow and a reference level.

For the case that two forces, 

are applied on the object A, from which the first force is twice in magnitude as the second and in an angle of 45 degrees between them, we would sketch:

Image 1:Two Forces on an Object

In the case that we want to design a vector vertically to the page, we design it, in the case its direction is outwards the page, as a circle with a dot inside it (representing the nose of the arrow) and in the case its direction is inwards the page, as a circle with an x into it (representing the rear part of the arrow).

 

Vector Geometric Addition

What if we want to calculate the total force that is applied on the object A. Vectors can be added. Actually we can apply on them all the operations. In order to add geometrically two vectors together, we can: 1) either put one vector at the end of the other and calculate the length of the straight line from the beginning to the end or 2) we can draw at the end of each vector a parallel line to the other vector and calculate the diagonal of the rhomb from the common point of the vectors to the end.

Image 2: Vectors Geometric Addition

From Image 2 we can see the magnitude and the direction of the total force applied on the object A. The magnitude of a vector can also be denoted as |vector|, for the example above is :

and it is a number.

 

Vector’s Geometric Subtraction

In order to subtract one vector from another, we just add to a vector the inverse of the other. To inverse a vector we rotate it 180 degrees.