PHILLIPS AND MOORE. — LINEAR DISTANCE AND ANGLE. 67 



The distance from a point to a line is zero if the point hes on the 

 hne or if the plane of the line and the point pass throngh F (assuming 

 that the point does not lie in and that the line does not belong to p). 

 Since the order of A and r is immaterial in the formula for Arwe. write 



Hence 



a r = 



A r = r A. 



17. Angle between line and plane. Dual considerations give 

 for the angle between a line r and a plane a, the expression 



— — i(t>o. r) 



(^>-= ra = -— — . . . (20) 



{F a) {p r) 



Let a be the plane at distance X from A and r the line all points 

 of which are at distance X from s. Then 



a = X (0 A) cf) — p A, 



r = X[</)SF]+ {ps) p-s. 



(0 {\{ct>A)ct>-pA] {\[cl> s F] + ip s) p - s]) 



{F {\(ct> A)<t>-pA\) (p {\<j>s F -^ (ps) p - s}) 



_ {-[pA]{{ps)F-sct>\) _ {ps){<t>A) + {pA)-{ct>s) 

 (0 A) {p s) icf> A) {p s) 



But [pA-scl>] = \p\{A<j>)s-A-scj>]], 



and {p[A-<l)s]] = {A-s-cjip) = (AsF). 



Hence a r = — - — - = .4 s. 



(.<t>A){ps) 



Therefore the angle between a line and plane is equal to the distance 

 between the line and point corresponding to them through the dis- 

 tance correlation. In particular for X = 0, we see that the distance 

 between point and line is ecjual to the angle between their polar plane 

 and point with respect to the complex p. 



18. Line Area of a triangle. We define the area of a triangle 



^ 5 C as a function A B C oi three points such that if the vertex is 

 fixed and the base moved along its line, the area is proportional to the 

 base. Hence if A is the vertex of the triangle and s the line on which 

 the base B C lies 



ABC = kB~C-A^, 



