LEWIS. — ON FOUR-DIMENSIONAL VECTOR ANALYSIS. 109 



If each factor has a subscript which the other has not, the inner 

 product is zero. 



The inner product of two identical unit vectors is equal to unity. 



In the remaining case, one factor having a higher order than the 

 other, the subscripts of the former should be transposed until those 

 subscripts occurring at the right are the same and in the same sequence 

 as in the factor of lower order. These common subscripts are then 

 cancelled and a unit vector with the remaining subscripts, in the se- 

 quence in which they stand, forms the inner product. Thus for 

 example, 



From these rules we obtain immediately the equations, 



ab = ^i-i^i + aobo + ^3^3. (10) 



aa == a^ r:r ai'^ + (72' + «3^ (11) 



AB = A,B, + A,D, + jUBs. (12) 



These products are scalars. On the other hand the product aA is a 1- 

 vector lying in the plane A and perpendicular to the projection of a 

 upon A, namely, 



aA = Aa = (^.12^2 + AisCh) K + (^l2i«i 4- ^-lasfl's) ^2 + 



(.-Isi^i + .132^1-2) k3. (13) 



Finally, the product of a pseudo-scalar and a 1 -vector is the perpendicu- 

 lar 2-vector, that of a pseudo-scalar and a 2-vector is the perpendicular 

 1 -vector, the product in each case having a magnitude equal to the 

 product of the magnitudes of the factors. 



The rules for outer multiplication may likewise be stated by stating 

 the rules for the unit vectors. 



K1XJC2 = K12 ; K1XK23 = K12XK3 = K123, 



kixki == ; kixki2 = ; K^^xk^o = ; kijxkig = ; (14) 



^1^^123 ^^ ■> ^12^^123 ^^ ; 1^123^'^123 ^^ 0. 



These statements may be geieralized and the following rules will hold 

 also for unit, mutually perpendicular, vectors in space of any dimen- 

 sions : 



If two unit vectors possess any subscript in common, their outer 

 product is zero. 



In all other cases the outer product is a unit vector having all the 

 subscripts of both factors, in the sequence in which they occur in the 

 factors. 



