18 



GEOMETRY OF VECTORS. 



[CHAP. ii. 



AC, of sense indicated by the order A, C, and of magnitude R, 



such that 



P:Q:R = AB:AD:AC, 



is equivalent to the two given vectors localised in AB, AD. 



Except in the case of parallel vectors this construction enables 

 us to replace a system of vectors localised in lines lying in a plane 

 by a single vector localised in a line in the plane. We can replace 

 two of the vectors by their resultant, then this resultant and a 

 third by their resultant, and so on. 



But the difficulty encountered when we seek to replace pairs 

 of parallel vectors can generally be removed, and the method by 

 which this is accomplished will at the same time furnish a method 

 for replacing a given set of localised vectors by simpler equivalent 

 sets of localised vectors which is of much more direct application 

 than the method just described. 



The principle on which this method rests is that as two equal 

 vectors of opposite senses localised in the same line are equivalent 

 to zero, any set of vectors localised in lines is equivalent to any 

 other set which differs from it only by containing, in addition to 

 the original vectors, pairs of equal and opposite vectors, the 

 vectors of any pair being localised in the same line. 



16. Moments. The moment about a line L of a vector 

 localised in another line L f is the product, with a certain sign, of 



Fig. 14. 



the magnitude of the resolved part of the vector at right angles to 

 the line L and the length of the common perpendicular to the 

 two lines. The rule of signs is as follows: One of the two senses 



