430 PROCEEDINGS OF THE AMERICAN ACADEMY. 



factor. Furthermore the outer product will always obey the associa- 

 tive law, and the inner product the commutative law. 



28. The inner product of any 1 -vector into itself may, by an im- 

 mediate generalization of the definition in plane geometry (§ 14), 

 be defined as ec{ual to the square of its interval, taken positively for 

 (7)-vectors, negatively for (6)-vectors. The inner product of two 

 1-vectors is equal to the inner product of either one and the projection 

 of the other upon it. The rules for the unit coordinate vectors are 

 therefore 



ki-ki = k2-k2 = 1, k4*k4 = — 1, ki'k2 = krki = k2«k4 = 0. 



The product of two vectors 



a = aiki + ojko + a4k4, b = 6iki + b'k-i + 64k4, 



is a'b = chbi -j- a_&2 — aibi. 



The inner product a* A of a l-vector and a 2-vector will be a 1-vector 

 in the plane A and perpendicular to a (that is, perpendicular to the 

 projection of a on A) ; its magnitude will be equal to the product of 

 the magnitude of A and the magnitude of the projection of a on A; 

 its sign is best determined analytically. If a and b are perpendicular 

 1-vectors we may make the convention 



(axb)'b = a(b'b), or (axb)«a = — b(a'a). (21) 



Thence follow the rules for the unit vectors, 



24 We may show that these rules do give an inner product which in all cases 

 agrees with the geometric definition above stated. 



The condition that a«A lies in tlie plane A is that the outer product of it 

 and A shall vanish, that is, (a'A)xA = 0; the condition that it is perpen- 

 dicular to a is that the inner product of it and a shall vanish, that is, 

 (a«Aj«a = 0. These two products are 



(a«AjxA = [{ao A\o — «4 ^4 14) 7^124 + («i ^12 + c^ Aot) Au 



- (aiAu + 02^24)^ 12] ki24 = 0, 



(a«A)«a = fli («2'1]2 - 04-4 14) - 02(01^4 12 + 04^424) + 04 (oi^u + 02^424) = 0, 



as required. It is also necessary to show that the component of a perpendi- 

 cular to A contributes nothing to the product a 'A, so that the component in 



