ADDITION AND SUBTRACTION OF VECTORS 429 



The pnrallelogram is completed by drawing P 2 P parallel to 

 OP 1( and PjP parallel to OP a . Then OP will represent the dif- 

 ference of the two vectors, or (A B), being the angle its line 

 of action makes with OX, and OP the magnitude. 



FIG. 141. 



208. The Vector Polygon. In order to find the sum of a system 

 of vectors the parallelogram law must be used time after time, 

 and this continued application of the parallelogram law gives rise 

 to the vector polygon. 



Let A, B, C, D . . . be a system of vectors whose magnitudes 

 are p v p 2 , p 3 , p 4 . . . and whose directions are Oj, 6 2 , 6 3 , 6 4 . . . 



Let OA, OB, OC, OD . . . make angles Q lt 2 , 6 3 , 6 4 . . . with 

 OX, and OA =-- Pl , OB = p 2 , OC = /> 3 , OD = / 4 . . . . (Fig. 142.) 



By completing the parallelogram OAaB, the diagonal Oa will 

 represent the sum of the vectors A and B. 



By completing the parallelogram OabC, the diagonal Ob will 

 represent the sum of the vectors Oa and C, that is the sum of the 

 vectors A, B, and C. 



Similarly Oc will represent the sum of the vectors A, B, C, and 

 D. It is evident that by drawing OA X parallel to OA, and making 

 OA 1 = OA = p 1 , by drawing A. l a l parallel to Aa and making 

 Aa^Aa = p z , Oaj will be exactly the same as Oa, and can there- 

 fore represent the sum of the vectors A and B. 



By drawing a^j parallel to ab, and making a^ = ab = p 3 , O6 1 

 will be exactly the same as Oft, and can therefore represent the 

 sum of the vectors A, B, and C. Similarly Ocj, being exactly 

 the same as Oc, will represent the sum of the vectors A, B, C, 

 and D. Thus OA^^Cj gives a polygon in which the arrows, 

 denoting the sense of each vector, follow each other round in 

 cyclic order ; Oc a is the closing line of this polygon. If this 



