6, 7] NUMBER. 11 



In order to avoid the inconvenience of the negative sign in 

 this, Hamilton's notation, we shall call the negative of the scalar 



product the geometric product and denote it by R^R.-,, so that 



( i ) RR 2 = X^ 2 + 7, F 2 4- Z,Z 2 = R,R 2 cos (R 1 R 2 ). 



If two vectors are perpendicular, their scalar product is zero. 



The vector' part R 3 of the product RA, or the vector product, 

 which will be denoted by the symbol VRA, has the components 



(2) X 3 = YiZ 2 Z l F 2 , Y 3 Z^K^ XZ. 2 , Z z = X l F 2 



It is to be noticed that the suffixes 1, 2, 3, appear in cyclic order, 

 as do the letters in the terms on the left and the first terms on the 

 right. If we multiply these equations by the corresponding com- 

 ponents of either R^ or R 2) we get identically, 



(3) RA = X,X S +Y,Y 3 + Z,Z 3 = 0, 



R 3 m X 2 X S + F 2 F 3 + Z 2 Z 3 = 0, 



showing that the vector product is perpendicular to each of the 

 vectors involved. Squaring and adding the equations (2), we get 



R* = X* + F 3 2 H- ^ = F^ 2 + ^ 2 F 2 2 + Z^ 



= (Xf + F, 



= R*R* (1 - cos 2 (JBA)) = RfRf sin 

 so that 

 (4) R 3 = | V^A | = RA sin 



The vector product of two vectors is accordingly perpendicular 

 to their plane and its tensor is equal to the product of their 

 tensors and the sine of their included angle, or geometrically, to 

 the area of the parallelogram having them as sides. 



The equations (i) and (2) show that 



