168 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



Multiplying any vector by a scalar, multiplies its magnitude by that 

 scalar. Multiplication of a vector by a scalar follows the laws of asso- 

 ciation, commutation and distribution. 



mSi. = mai'k.i + ma^2. + ma^z (8) 



In multiplying one vector by another two kinds of product are to be 

 distinguished, which, following Grassmann, we shall call the inner and 

 outer products,^ and define as follows. 



The inner product follows the distributive and commutative laws. 

 It is in general a vector and its order is the difference between the 

 orders of the factors. (Thus the inner product of two 1 -vectors is a 

 0-vector, or scalar, the inner product of a 1- vector and a 2-vector is a 

 1 -vector.) 



The outer product follows the distributive and associative laws. It 

 is a vector of which the order is the sum of the orders of the factors. 

 (Thus the outer product of two 1 -vectors is a 2-vector, that of a 1- 

 vector and a 2-vector is a 3-vector, which in three-dimensional space 

 is a pseudo-scalar.) 



The inner product of two vectors will be indicated merely by their 

 juxtaposition, for example, ab ; AB ; a A. 



The outer product will be indicated by a cross ^ placed between two 

 vectors, for example, axb ; AxB ; axA. 



Since both kinds of products follow the distributive law they may be 

 completely defined by the rules governing the multiplication of the 

 simple unit vectors. The rules for inner multiplication are as follows, 



kiki = 1 ; kiko = koki = 



^12^12 =^ 1 ', ki2ki3 = ki3ki2 = 



■^123*^123 ^ 1 '} 



K2K12 = Kl2^2 '^^ ''^V ', kik23 = k23ki =: 



•^23^123 =^ ^123^23 =■ S-l ', 



These statements may be generalized, and the following rules will hold 

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

 sions : 



^ The terms scalar product and vector product would obviously be mis- 

 nomers in the present system. 



" This symbol for the outer product is used by Gibbs and his followers 

 (see Gibbs, Collected Papers; Wilson-Gibbs, Vector Analysis: Coffin, Vector 

 Analysis), and has several advantages over the more awkward square brackets 

 fa, b] frequently used to express the outer product. The brackets were used 

 by Grassmann, but had a far more Rcneral significance than the product de- 

 fined above. (Ausdebnungslehre von 1862, p. 28.) 



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