to JS^on- Metrical Vector Algebra. 139 ? 



(i. e. not exceeding 7r) made by a, b. Then we call vector- 

 product of A into B and denote by 



C = VAB, 



a third vector whose tensor is C—AB sin 0, drawn perpen- 

 dicularly to a, b and so that a right-handed rotation about 

 C carries the vector a to b. 



From this definition it follows at once that 



VAB=-VBA=JJ?Vab. 



Further, i£ A, B are coterminal vectors, VAB=:0 (corre- 

 sponding to the common case of u parallel" vectors). In 

 particular, for our normal, right-handed triad of unit vectors 



i, J, k 



Vii=0, etc. Vij=k, Vjk=i, Vki=j. 



The distributivity of this vector product can be proved in a 

 variety of ways. The shortest and easiest seems to be the 

 following proof. Let 



A^Aii + A.j + Agk, 



and similarly for B and C. Then, C being normal to A, B, 

 by definition, we have 



CA^CiAi + 2 A 2 -f C 3 A 3 = 0, 



CB = OjBi + C 2 Bo + C3B3 = 0, 



c c 



whence, solving these two equations for the ratios ,— , jr y 

 we find at once 3 3 



C 1 =\(A 2 B 3 -A 3 B 2 ) J C 2 =X(A 3 B 1 -A 1 B 3 ), 



C 3 =X(A 1 B 2 -A 2 B 1 ), .... (18') 

 where A, is a scalar. Remembering that, by definition, 

 ( ? = CV 4- C? + C 3 2 = A 2 B 2 sin 2 6>, 



we have \ 2 =1, that is \= +1. To decide the sign take, for 

 instance, A = i, B=j. Then, by definition, C = k, /. e. 

 C l = C 2 = 0, C 3 = l; and since A 1 = B 2 =1, while A 2 , B u etc. 

 all vanish, we have, by (18'), X=l. Thus the expanded 

 form of the vector product (defined intrinsically at the 

 outset) is 



YAB==^A a B l -A 8 B 8 )+J(AjB 1 --AiBg) + k(A 1 B > -A 8 B 1 ),(18) 



familiar from ordinary vector algebra. Now, if M be any 

 vector, the components of B-fM along i, j. k are (by the 



