I895-] BAKER — DIRECTED MAGNITUDES. 



167 



to the location ol p„. But this amounts to a transference, a vector. 

 As this operation results in the formation of a new quantity, the vector 

 between the two points, the operation is called combinatory midtiplica- 

 tw7i to distinguish it from algebraic multiplication of weights. This 

 gives us the definition : — A combinatory or geometrical product of two 

 quantities not having an element of their extension in common, is the 

 geometrical quantity produced by the guided factor as it moves over 

 a path determined by the guidiiig factor. 



The guided factor is written f^rst. Thus, p^p^ is the posited 

 vector whose magnitude, direction and location of extension are 

 determined by the movement of/, X.o p„. Movement in the opposite 

 direction would, of course, have a different sign, ox p^p^ — —p^p^, 

 giving us the alternative law of multiplication. Evidendy/,/, =0, 

 or the prodicct of two identical factors is zero. 



Since the only properties of a vector are magnitude and direction, 

 Its guiding influence on a point must be to make it move a distance 

 and direction determined by the vector, and/, e (s denoting a vector) 

 becomes a vector through a fixed point. Hence -.— Guiding a point 

 by a vector converts the point into a posited or point vector, or locates the 

 vector. 



Guiding a vector by the location of a point locates the vector 

 through the point. Guiding the magnitude of extension of the vector 

 by the zero magnitude of the point must be just the opposite of guid- 

 ing the zero magnitude of the point by the magnitude of the vector 

 and hence must reverse the vector. Or, guiding a vector by a point 

 locates the vector and reverses its direction. Thus, />, £ = ep . 



Similarly \\\e product of two vectors is the plane area generated by 

 the movement of the first vector as it is guided by the characteristics 

 (magnitude and direction) of the second vector. As the angle between 

 the two vectors passes through tt the area passes through zero. 

 Hence -.— The sense of the area is + or — according as the guided 

 vector is guided by a vector on its left or right, or vice versa. Thus, 

 £■£. = — £,£,. 



Guiding a point by the characteristics of a point vector (posited 

 vector), magnitude, direction and location of extension, must result 

 in a posited plane area double that of the triangle determined by the 

 point and the posited vector. Denoting the posited vector by p„p^, we 

 have for the resulting ^rez p^p„_p^. Arranging the products 'of' the 

 three points in all possible ways, we find that the only way to separate 



?i, Proc. Roch. Acad, of Sc, Vol. 3, December, i8qq. 



