WILSOX AM) I.KW IS. — UKL.VTIN ITV. 431 



The inner product of two 2-vectors is a scalar w liicli is eciual to the 

 inner prothict of either vector by the projection of the other upon it. 

 The inner product of two perpencHcular 2-veetors is zero. The inner 

 product of a 2-vector by itself is luuiierically equal to the square of 

 its magnitude, and is positive in sign if the vector is of class (7), 

 negative if of class (5), Hence we have as rules of inner multiplication 

 for 2-\ectors 



ki2*kio = 1, ku'ku = kj4«k24 = — 1, 



kio'ki4 = ki2'k24 = ki4«k24 = 0, 



A- A = Avr — An- — Au', A-B = .412^12 — AuB^ — A.iB-u. 



29. Every 1-vector a, or 2-vector A in a three-dimensional space 

 uniquely determines, except for sign, another vector (respectively 

 a 2-vector or l-\ector) perpendicular to it and of equal magnitude. 

 This vector will be called the complement of the given vector, and 

 designated as a* or A* respectively. To specify the sign, the comple- 

 ment may be defined as the inner product of the vector a or A and the 

 unit 3-vector or pseudo-scalar ki24, where the laws of this inner product 

 are 



krki24 = k24, k,-ki24 = — ku, k4'ki.:4 = — ki2, 



kl2*ki24 = k}, ki4«ki24 = k2, k.4'ki24 = — ki. 



Thus 



a* = (oiki + aoki + a4k4)«ki24 = — • a^^yi — a^u + aik24, 



A* = (.'Ii2ki2 + ^]4kl4 + .'I24k24)'ki24 = — ^24kl + Ay^y + Ay^S.^. 



These complements satisfy the condition of perpendicularity pre- 

 viously derived (footnote 24), and the inner products 



a*, a* = 04- — Or^ — a^, a«a = or + a? — 04", 



A*-A* = ^24- + ^14- — A,o}, A-A = .4i2- — Au~ — ^424= 



the plane is alone of importance. We shall do this by deriving the expression 

 for a vector perpendicular to the plane A. Let 



c = ci ki + C2 k) + f'4 k4, n = /;i ki + i^yi-i + "4 k4 



be respectively any vector in the plane A and a vector perpendicular to the 

 plane. Then the products 



CxA = {CiA-n — C2^14 + C4-4l2)ki24 = 0, C«n = Ci ?(i + ^2 "2 — £"4 "4 = 



vanish. Hence it follows that the condition of perpendicularity for the vectors 

 n and A is 



n\: 112. «4 = -4241 — '•1]4: — An, 



and that n must be some multiple of -424ki - Juko - .4i2k4. Hy llie rules, 

 the inner product of this vector and A vanishes. 



