22 KEST AND MOTION 



where the summation is now taken over the various points in space 

 at which the original points are accumulated. Calling these points 

 in space A, B, C,--, the point x, y, % is said to be the centroid of the 

 points A, B, C, , corresponding to the multipliers m a , m b) m c> 

 By means of these results, equations (6) are reducible to 



X=x--j.m r> r=y.V r> Z=z-^.m r . (9) 



1 1 1 



Hence the resultant of the above set of vectors is directed along 



n 



the line 0(7, and its magnitude is 0(7- Vra r . As denned by equa- 



i 

 tions (9), the multipliers m r can be any numbers whatever, positive 



n 



or negative, so that the sum Vm r may be positive, zero, or nega- 

 tive. In particular, when the vectors are represented in magnitude 

 as well as direction by A^ , OA n) the resultant is directed along 

 OC oy and its magnitude is n OC 0) where n is the number of vectors 

 and the point C is the centroid, as denned above. Thus we have 

 the theorem : 



THEOREM. If vectors of magnitude m^OA^ m 2 OA 2 , act along 

 the lines OA lt OA 2 , , then their resultant is of magnitude 



(m l + ra 2 H )OG, and acts along OG, where G is the centroid of 



the points A v A 2 , - for the multipliers m l} m 2 , 



Obviously the parallelogram law is a particular case of this 

 theorem. 



EXAMPLES 



1. Find the resultant of two vectors of magnitudes 5P and 12 P which meet 

 at right angles. 



2. A vector P is the resultant of two vectors which make angles of 30 and 

 45 with it on opposite sides. How large are the latter vectors ? 



3. Show how to determine the directions of two vectors of given magnitude 

 so that their resultant shall be of given magnitude and direction. When is this 

 impossible ? 



4. Show that if the angle at which two given vectors are inclined to each 

 other is increased, their resultant is diminished. 



5. Under what conditions will the resultant of a system of vectors of magni- 

 tudes 7, 24, and 25 be equal to zero ? 



