408 PROCEEDINGS OF IHE AMERICAN ACADEMY. 



The inner product of any two vectors is equal to the inner pmrluct 

 of either one by the projection of the other along it. For either 

 vector may be resolved into two vectors one of which is parallel and 

 the other perpendicular to the other vector. Thus b may be written 

 as na + a', where ??a is the projection of b on a, and a' is perpendicu- 

 lar to a. Therefore 



b-a = 7ja,-a, -\- a'-a = ??a*a, 



which was to be proved. Geometrically the only puzzling case is that 

 in which the vectors are of different classes. Let OA (Figure 12) be 



a vector of class (7) and OB of 



_ class (5). The projections of 



X .^' ^^\ ^ OA on OB and of OB on OA 



/''"x^^s^----.^ ^^-•''' are respectively OB' and OA'. 



/ ^"-Ov "^^^r'^ Note that whereas 05' extends 



/ ,--^^*^ — ' ^ (y) ^^ ^^^ same direction as OB, 



/-'--' " ,-''' ^^^^ the vector OA' extends along 



•^ ,,^'' ^^^^ the opposite direction to OA. 



^^ Thus 05' is a positive multiple 

 ^^^^^'^ ^^- of OB, whereas OA' is a nega- 



tive multiple of OA. But the 

 inner product of OB by itself is negative, since the vector is of class 

 (5), while the inner product of OA by itself is positive, since the vector 

 is of class (7). Hence the inner product of OA and OB has the same 

 sign, whichever way the projection is taken. 



In obtaining the inner product of a singular and a non-singular 

 vector by projecting one upon the other, it is necessary to project the 

 singular vector upon the non-singular vector; for it is impossible to 

 make a perpendicular projection upon a singular vector. In case 

 both vectors are singular the method of perpendicular projection fails 

 entirely, and we must use analytical methods (or have recourse to 

 parallel projection). 



15. It will often be convenient to select two mutually perpendicular 

 lines as axes of reference. We will denote ^^ by ki and k4 unit vectors 

 along such axes, ki being the vector of the (7)-class, and k4 of class (5). 

 For these vectors we have the rules of multiplication 



ki'ki = 1, k4«k4 = — 1, ki«k4 = k4«ki = 0. 



18 We reserve the symbols k2 and ks for other unit vectors of class (>) in 

 space of higlier dimensions. 



