l895-] RAKER — DIRECTED MAGNITUDES. 169 



Thus, p„p^. f>^p„=p^^. p^^p^p^ = _p^ .p^^p^p^^_p p_^. pj^ 



that IS :— The product of two coplanar posited vectors is not commutative. 

 This product introduces a new result. Out of the product of the 

 two posited vectors, each having magnitude, direction and location of 

 extension, we get, as shown above, a point /„, the common element of 

 the posited vectors, and a scalar area p„p,p,, that is, the two factors 

 have produced a quantity of a lower order than either of the factors. 

 This is called a regressive product. 



We might have expected this, for out of the combination of the 

 two posited vectors, the common element of extension, the point in 

 this case, is the only quantity that has any distinctive geometric char- 

 acteristic, the area generated being a mere scalar, differing from all 

 other areas in magnitude only, and the product of the two posited 

 vectors is a weighted point. 



The same considerations apply to the product of a posited vector 

 and a posited area, etc., thus suggesting the rn\e :— The product of 

 two posited quantities which have a common element of extension is that 

 element multiplied by soine scalar. 



Guidmg a \'ector by two others in succession gives the parallel- 

 opiped having the vectors for its edges. The product of four points 

 is in a similar manner the parallelopiped six times the connecting 

 tetrahedron. Arranging the products of the four points in the possible 

 different ways, taking into account the association into single, double 

 and triple factors, we find that the only basis of division into two 

 classes is the rule -.—Lookiiig from {or along) the operator if the 

 points of the operator {or operand ) run in cyclical order the result is 

 unchanged. ___— — — t::^/^ 



Hence, Hj 



P.P.P-J,=P,P,P, -p, = Pp^ 

 =P.P. -P,P.=-L^L^ 

 =P.P. •P,P, = L,L^ 

 = — A ■P.P.P, = —P, 



P< ■ P,P,P, IS negative because viewed from p^, p,p„p, runs clock- 

 wise, whereas ^*hen p^ is viewed from p^p^p^, p^p„p^ runs counter 

 clockwise. Henee -.-The product of a point and a posited plane 

 vector (P) is not cofnmutative ; the product of two dip lanar posited 

 vectors is commutative. 



