434 PRACTICAL MATHEMATICS 



Thus in space, a vector is specified completely by 



(1) O, the point of application. 



(2) The line of action which makes angles a, (3, and y with 



the axes OX, OY, and OZ respectively. 



(3) A length OP measured from O along the line of action 



and equal to p, the magnitude of the vector. 



(4) The sense ; this is positive if the vector acts away from 



O and negative if the vector acts towards O. 



Only two angles need be given since the third can always be 

 found from the relation cos 2 a + cos 2 ^ + cos 2 y = 1. 



Thus in dealing with a system of vectors in space, having a 

 common point of application, 



If Pi> P-z> Ps are the magnitudes 



<*!, a 2 , oc 3 . . . the angles made with the axis of x 



or l lt 1 2 , 1 3 ... the corresponding direction cosines 



Pi (^2> Pa the angles made with the axis of y 



or TTij, w 2 , m s . . . the corresponding direction cosines 



Yi Yz> Yi th e an gl es made with the axis of z 



or n lf n 2 , n 3 . . . the corresponding direction cosines 



The components in the direction OX are p t cos a 1( p 2 cos a 2 , 

 p 3 cos a 3 . . . and since all these have the same line of action they 

 can be reduced to one vector, of magnitude X, in the direction OX, 



and X = p i cos <x. l + p z cos <x 2 + p a cos oc 3 . . . 



Similarly, resolving the vectors in the direction OY, and if Y 

 is the algebraic sum of the components in that direction, 



then Y = p x cos (^ + p. 2 cos (3 2 + p 3 cos (3 3 . . . 



Also resolving the vectors in the direction OZ, and if Z is the 

 algebraic sum of the components in that direction, 



then Z = /Oj cos -^ 1 + p 2 cos y2 + Ps cos Ys 



Thus the whole system can be reduced to one of three vectors 

 whose magnitudes are X, Y, Z, and whose lines of action are the 

 three axes of reference, OX, OY, and OZ. If p is the magnitude 

 of the vector sum, and a, p, and y the angles it makes with the 

 axes of reference, 



