THE SCALAR PRODUCT OF TWO VECTORS 437 



The scalar product = ON X x ON a 



= Pl p 2 cos (0! - 6) cos (6 a - 0) 



>s (0 a + X - 20) + cos (0 2 - 



This is evidently a maximum when 



cos (O x + 2 - 20) = 1 



or 



That is when the line OP bisects the angle between the vectors 



The maximum value 



- PlP2 {l + COS (02- 



PiPz cos 



2 - 0! 





FIG. 144. 



Also if = : , that is, when OP coincides with the line of action 

 of the first vector, 



the scalar product = - p^p 2 {cos (0 2 a ) -f cos (0 a 



COS (0 2 - 0J 



and if = 2 , that is, when OP coincides with the line of action of 

 the second vector, 



the scalar product = - Pjp 2 {cos (0 X 2 ) + cos (0 2 0J } 



a 

 = ptf z COS (0 2 - 0j) 



Hence if the scalar product is taken in the direction of one or 

 other of the two vectors, it becomes the " product of the magni- 

 tudes of the vectors and the cosine of the angle between them." 

 This is the definition of the scalar product as applied to actual 

 practice. 



