2022] 



PARALLEL VECTORS. 



23 



It follows that the set of four vectors P, P, and Q, Q are equi- 

 valent to zero. 



This theorem shows that a couple may be replaced by any 

 other couple of the same moment and sense and in the same plane. 



Before going on to Theorem II. we must prove a general 

 theorem concerning the resultant of two vectors localised in 

 parallel lines but not forming a couple. This theorem might be 

 proved independently of Theorem I., but the proof of it becomes 

 very much simplified if Theorem I. is assumed. 



22. Vectors localised in parallel lines. Let P, Q be the 



magnitudes of two vectors localised in parallel lines, A and B any 

 points on these lines, d the distance between the lines. 



Fig. 19. 



In the line in which P is localised introduce two vectors each 

 of magnitude Q and in opposite senses. Then the pair of vectors 

 is equivalent to a couple of moment Qd and a single vector local- 

 ised in the same line as P. The magnitude of the single vector 

 is P + Q (Fig. 19, a) if P and Q have like senses, and is P ~ Q 

 (Fig. 19, ft) if P and Q have opposite senses, and its sense is that 

 of P or Q when these have like senses, but when P and Q have 

 opposite senses it is that of the vector P if P > Q, that of the 

 vector Q if Q > P. In either case let the magnitude of the single 

 vector be R, and in the second case, to fix ideas, take Q>P. 



Replace the couple of moment Qd by a couple in the same 

 plane consisting of two vectors each of magnitude R, one of them 

 being localised in the same line as P and in the opposite direction 

 to R, and the other in a parallel line at a distance Qd/R on that 



