CHAPTER II. 



GEOMETRY OF VECTORS. 



7. Definition of a Vector. A vector is an object of mathe- 

 matical reasoning which requires for its determination (1) a number 

 called the magnitude of the vector, (2) the direction of a line 

 called the direction of the vector, (3) the sense in which the line 

 is supposed drawn from one of its points, called the sense of the 

 vector, and which obeys a certain rule of mathematical operation 

 to be presently stated. 



Let any particular length be taken as unit of length. Then 

 from any point a straight line can be drawn to represent the 

 vector* in magnitude, direction, and sense. The sense of the line 

 is indicated when two of its points are named in the order in which 

 they are arrived at by a point describing the line. 



The rule of mathematical operation to which vectors are 

 subject is a rule for replacing one vector by other vectors to 

 which it is (by definition) equivalent. 



This rule may be divided into two parts and stated as 

 follows : 



(1) Vectors represented by equal and parallel lines drawn 

 from different points in like senses are equivalent. 



(2) The vector represented by a line AC is equivalent to the 

 vectors represented by the lines AB, BC, the points A, B, C being 

 any points whatever. 



* The line is not the vector. The line possesses a quality, described as 

 extension in space, which the vector may not have. From our complete idea of the 

 line this quality must be abstracted before the vector is arrived at. On the other 

 hand the vector is subject to a rule of operation to which a line can only be 

 subjected by means of an arbitrary convention. 



