The Imaginary in Geometry 



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and is perpendicular to y = ix through the origin. I or (00, ice) 

 Y 



i(y-i) 



f / M 



0* 



t 



Fig. 23. 

 is their common point the distance of which from (2, i) is 



LV(-i- — 2) 2 — (.r-i) 2 = LV-3-f + 3 : 



coz 



When the line is of real slope, the perpendicular to it through 

 any element, i. e., any red vector, also has real slope. The distance 

 has for its real part the distance from the black point of the red 

 vector to the black line of the line of real slope while the imagi- 

 nary part will be the difference of the projections upon the per- 

 pendicular of the given red vector and the red vector of the inter- 

 section. Thus from y=i to (x, y) the distance is 



+ / + »(?" — I). 



The distance from one red vector to another can be represented 

 in terms of the distances along two lines of real slope, one through 

 each vector. Thus in fig. 24, 



AC 2 =AB- + BC- + 2AB ■ AC cos u 



= AE 2 + EC- -f 2AE • EC cos <f>. 



In the last member EC is a pure imaginary = ik say. Taking 

 coordinates as indicated. 



23 



