24, 25] 



EQUIVALENT SYSTEMS. 



27 



directions, and senses of two lines E B and B C . This is the 

 vector law. 



25. System of localised vectors in a plane. Let a vector 

 of any magnitude P be localised in a line AB, and let be any 

 point not in the line AB. Through draw a line 

 parallel to AB, and let there be two vectors each 

 of magnitude P and of opposite senses localised 

 in this line. Then the system of vectors is equiva 

 lent to a vector localised in the line through 

 parallel to AB, of magnitude P, and having the 

 sense of the original vector in AB, and a couple of 

 moment Pp, where p is the distance of AB from 

 0, and of a definite sense, with its axis perpen 

 dicular to the plane A OB. 



Any given system of vectors in a plane can in 

 this way be replaced by a vector localised at a point 

 in the plane, which is the resultant of equal parallel vectors in 

 the senses of the given vectors, but localised in lines through 0, 

 and a couple whose axis is perpendicular to the plane and whose 

 moment is 2 (+ Pp), where P is the magnitude of any one of the 

 original vectors, p the perpendicular on its line from 0, and the 

 sign of each term is determinate. 



Let R be the resultant of the vectors at 0, and G the moment 

 of the couple. If R is not zero replace G 

 by two localised vectors, each of magnitude 

 R, one localised in the line of R through 

 and in the sense opposite to R, and the 

 other in a parallel line at a distance G/R 

 from 0. The whole system is then equi 

 valent to this last vector. 



Fig. 23. 



If R is zero the whole system is equi 

 valent to the couple G. 



If R and G are both zero the system is 

 equivalent to zero. 



Thus any system of vectors localised in 

 lines lying in a plane is equivalent to a 

 single vector localised in a line lying in the 



O/R 



Fig. 24. 



