PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 783 



From the fact that segments are directed we see that the product of 

 two points obeys the alternative law 



AB = - BA. (29) 



Also when the two points coincide the segment is zero, that is, 



AA = (30) 



On the other hand, if 



AB = 0, 



either the magnitude of one of the points is zero or the segment is 

 zero. Then if neither point is a zero point and the product vanishes, 

 the two points will coincide. Here again it is well to note that the 

 product of two points on a line through F does not necessarily vanish 

 although the segment is of zero length or magnitude. 



41. We have used the notation 



ABC 



to represent the triangle determined by the three points. We shall 

 consider this also as the product of the three unit points and take it to 

 mean the line area of the triangle determined by the three points. If 

 the points are not unit points, the product will mean the area multi- 

 plied by the product of the magnitudes of the points. From this defini- 

 tion we see that the product vanishes if the three points are on a line, 

 neither point being on f. Conversely, if the points are not zero points 

 and the product vanishes, they are on a line. Therefore 



ABC = 



is a necessary and sufficient condition that the three points be linearly 

 related, i. e. collinear. 



From the fact that area is directed we see at once that 



ABC = BCA = CAB = - ACB = - CBA = (31) 



42. From duality we now define the product of two lines as the 

 sector determined by the two lines multiplied by the product of the 

 magnitudes of the lines. When the two lines coincide the sector 

 vanishes and we have as for points 



ab = 0. 



Conversely, if the lines are not zero lines, ab = is the condition that 

 the two lines should coincide. In the case of two lines as in the case 



