432 



PRACTICAL MATHEMATICS 



It should be noticed that in the second case the sense of C is 

 changed, while in the third case the senses of B and D are changed, 

 in their respective vector polygons. 



209. Addition and Subtraction of Vectors by Resolution. If two 

 vectors have the same line of action, their magnitudes can be added 

 or subtracted according as their senses are like or unlike. This 

 enables us to add and subtract vectors by resolving them in the 

 horizontal and vertical directions, and then finding the algebraic 

 sum of their horizontal and vertical components. Referring to 

 the system of vectors in the previous paragraph, the horizontal 

 components are /Oj cos X , p z cos 2 , p 3 cos 3 , /o 4 cos 4 . . . and 

 since all these have the same line of action they can be reduced 

 to one vector, of magnitude H, in the horizontal direction 



and H = 



cos 



-f p 2 cos 2 -f- p 3 cos 3 + p 4 cos 4 + 



The vertical components are p t sin Q lt p 2 sin 2 , p 3 sin 3 , p 4 sin 4 

 . . . and since all these have the same line of action, they can be 

 reduced to one vector, of magnitude V, in the vertical direction 



and V = 



sn 



sn 



+ 



sn 



2 + /o 3 sin 



Thus the whole system is reduced to the sum of two vectors : 

 one, of magnitude H, acting horizontally ; the other, of magni- 

 tude V, acting vertically. On applying the parallelogram law, 

 the parallelogram becomes a rectangle. 



Then 

 and 



where p is the magnitude and the direction of the resulting sum 

 of all the vectors. 



If one of the vectors is to be subtracted, then the algebraic signs 

 of its horizontal and vertical components must be changed. Tak- 

 ing the same example as in the previous paragraph and working 

 it in this manner 



