PHILLIPS AND MOORE. 



ALGEBRA OF PROJECTIVE GEOMETRY. 



757 



from Q upon RF and then from P upon AB. Vectors on AB have 

 therefore the same ratios as vectors on CD. Since distance has been 

 defined as proportional to the vector along CD, it is proportional to the 

 vector along any other line AB. 



20. It has been mentioned that two vectors on different lines may 

 be equal though their lengths are not equal. This is shown in 

 Figure 10. Let AC and BD intersect on f at P and let AB, CD cut 

 PF in M, N. The vectors AB and CD are equal. Suppose now 



Figure 10. 



Then, by proportionality 



and 



Also since NA = NC, 



AB = CD. 



AM = CN, 



BM = DN. 



AM = AN. 



Consequently P must be on the line AF. For the same reason B lies on 

 the line FP. The vectors AB and CD are then zero. Non-vanishing 

 equal vectors on different lines can not have the same length unless the 

 point F lies on the line f 



21. Summary. In this section we have found a species of linear 

 distance defined by a point F and line f The distance between any 

 two points on a line through F (but not on f) is zero. The distance 

 between a point on a line through F and the point in which that line 

 cuts f is indeterminate. The distance between two distinct points of a 

 line not passing through F is finite and not zero but becomes infinite 



