WILSON AND LEWIS. — REL.\.TIVITY. 409 



Any two vectors a and b'niay be written in the form 



a = rtiki + iiik-i, b = 6iki + 64k,, 



and tlie inner product is then, by the (Hstributive hiw, 



a*b = aibi — 0464. 



In terms of these unit vectors we may also express outer products. 

 If we write, for brevity, ku = ki x k^, the rules for outer multiplica- 

 tion are 



ki4 = — k4i, kii = k44 = 0. 



The outer product of the vectors a and b is therefore 



axb = ((hbi — 046]) ki4. 



Since ku represents a parallelogram of unit area, the question 

 arises as to why we write kixki as ku and not simply kjxki = 1. The 

 answer is that the outer product axb possesses a certain dimension- 

 ality, which, it is true, is not exhibited in a marked degree until we 

 proceed into a space of higher dimensions, but which renders it un- 

 desirable to regard the outer product as merely a scalar. We may call 

 it a pseudo-scalar, and later extend this designation to 7i-dimensional 

 figures in a manifold of n dimensions. 



Every vector in two dimensional space uniquely determines, except 

 for sign, another vector, namely, the one equal in interval and per- 

 pendicular to the first. This vector will be called the complement of 

 the given ^•ector. To specif}' this sign, the complement a* of the 

 vector a maybe defined as the inner product of a and the unit pseudo- 

 scalar ku, namely, a* = a'ku, where the laws of this inner product are 



ki-ki4 = — k4, k4-ki4 = — ki. . 



Thus if a = Oiki + atk-t, then for the complement 



a* = (oiki + a4k4)* = (aiki + a4k4)'ki4 = — a4ki — «ik4. 



This type of multiplication, as will be seen later, obeys all the general 

 laws of inner products (§§ 27, 29). 



Referred to a set of perpendicular unit vectors, the singular vectors 

 take the form 7i(± ki ± k4). The complement of a singular vector is 



n(± ki ± k4)*=/i(± ki ± k4)'ki4 = 7i(=f k4 =f ki), 



that is, the complement of a singular vector is its own negative. 



