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macfarlane: algebra of physics 



Consider next the product of a quadrantal versor and a unit- 

 vector. From the diagram it is evident that h j = k but 

 jir/2 ^ _ _ j, . a j gQ yjw/2 ^ _ ^ Hence we have the table : 



Consider finally the product of two unit-vectors. From consid- 

 erations of projection I derived the table 



This table is implicitly assumed by vector-analysts; but there is 

 an evident difficulty in reconciling it with the principle of dimen- 

 sions and in harmonizing it with the products of quadrants, 

 which are in a sense imaginary unit vectors. To overcome the 

 former difficulty I adopted the following theory (p. 79) : 



. In such an expression as xi it is more philosophical and correct to 

 consider the x as embodying the unit, while i denotes simply the axis. 

 I look upon the magnitude as containing the physical unit, to be arith- 

 metical ratio and unit combined, and different vectors have different 

 physical units. An axis is not a physical quantity but merely a direc- 

 tion. 



In pursuance of this theory I considered the types of compound 



