I -S 3.03 



to Non-Metrical Vector Algebra, 125* 



It' m, /* be any scalars, then | mA | =m [A \ , | nB | = « | B |, so 

 that, by the definition (6), the product of mk. into nB is 

 equal to mn times (AB). In view of this property we can 

 henceforth drop the brackets and write the scalar product 

 simply AB and, instead of ((>), 



AB=[A|.jB|ab (6a) 



If A, B are orthogonal, i. e. if ab = 0, we have AB = 

 unless, of course, A or B are infinite or T- vectors. 



We have already seen that aa=l, for any unit vector a; 

 instead of this we shall write a 2 = l. Thus, for any vector A, 



A 2 =|A| 2 = A 2 (7> 



In words, the scalar autoproduct of any vector is equal to 

 the square of its tensor, exactly as in ordinary, euclidean, 

 vector algebra. 



But what makes our generalized scalar multiplication of 

 vectors a powerful operation is the fact that it shares with 

 the ordinary one the property of distributivity . This can be 

 proved in a variety of ways. The shortest of these seems to 

 be the following one. 



Fig-. 4. 



N.'< B 



By (6) or (6a) the definition of AB is reduced to that 

 of ah, given originally by (1). In the preceding Section we 

 have already given a geometrical representation of this 

 number. Another, still simpler representation is obtained 

 bv drawing from b the perpendicular to a. Let n (tig. 4) be 



