406 PROCEEDINGS OF THE AMERICAN ACADEMY. 



area does not arise from any positive or negative geometric charac- 

 teristics of the area itself, but from an interpretation or convention 

 concerning the way in which one area is considered as generated 

 relative to another, and is required for analytic work. We shall make 

 the convention that a^b and ( — a)xb or ax(— b) have opposite signs. 

 The outer product of a vector by itself or by any parallel vector is 

 zero, because the parallelogram determined by these vectors has zero 

 area; thus axa = 0. The associative law for a scalar factor is valid, 

 because multiplying one side of a parallelogram by a number multi- 

 plies the area by that number; thus 



(na)xb = naxb = ax(nb). 



The distributive laws, 



ax(b + c) = axb + axe, (a + b)xc = axe + bxe, 



also hold; for inspection shows that the parallelogram ax(b + e) is 

 equal to axb plus axe. The anti-commutative law, 



axb = — bxa, 



holds; for 



(a + b)xfa + b) = axa + axb + bxa -f bxb = 0. 

 Hence 



axb = — bxa. 



14. Thus far we have proceeded by means of the parallel-trans- 

 formation alone. It is evident that this much of vector algebra is 

 common to all geometries, including the Euclidean and our non- 

 Euclidean geometry, in which there is such a parallel-transformation. 

 The other type of product, the inner product, cannot be defined with- 

 out some concept of rotation or perpendicularity, or its equivalent. 



We shall so define this inner product a-b that it obeys the associa- 

 tive law for a scalar factor and the distributive and commutative laws, 

 namely, 



(wa)«b = 7?a«b =a'(nb), 



a«(b + c) = a.b + a.c, 



a«b = b«a, 



and furthermore remains invariant during rotation. 



As the fixed lines are fundamental in rotation it is sometimes ex- 

 pedient to resolve vectors into components along these directions. 

 Let p and q be definite vectors in the two fixed lines; any vector in 



