2, 3, 4] SCALARS AND VECTORS. RESULTANT. 



We have from 4) and 5) 



cos I = = x j 



cos 11= = 



COS 



Also multiplying the equations 4) respectively by cos %, cos /i, cos v, 

 and adding, 



7) s x cos k -\- s y cos /i + s z cos v = s. 



Whatever quantities are needed to completely specify a quantity 

 are called its coordinates. A point has three, and we have seen 

 that a vector also has three, which may be taken as s x , s y , s z . In 

 this sense all vectors are to be considered as equal whose lengths are 

 equal and directions parallel irrespective of the absolute positions of 

 their ends. It is, however, sometimes necessary to distinguish vectors 

 equal in this sense, but whose ends do not respectively coincide. 

 To determine such a vector we must know not only its length and 

 direction, but also the position of one end. It will therefore be 

 specified by six coordinates, which may be the three coordinates 

 of one end, x ly y^, # , with the projections, s x , s y , s z , or the co- 

 ordinates of both ends, x if y 19 lf % 2 > 2/2; #2- ^ n an J case there will 

 be six coordinates. Such a vector may be called a fixed vector to 

 distinguish it from the ordinary or free vector. 



4. Addition of Vectors. To add two vectors means to take 

 successively the steps denoted by them, their sum being a single step 

 equivalent thereto. For example, (Fig. 1) 



The vectors AB and BC are called 

 the components of AC, which is called 

 their resultant, or geometrical sum. 



We may state the rule: Place the 

 initial point of the second vector at the 

 terminal point of the first, the resultant or 

 geometrical sum is the vector from the 

 initial point of the first component to the 

 terminal point of the second. This con- 

 struction gives us the so-called triangle of 



vectors. By continuing the process any number of vectors may 

 be added, giving us the polygon of vectors. 



Fig. 1. 



