70 PRACTICAL MATHEMATICS 



45. The Plane. The plane is represented by the general equa- 

 tion of the first degree in x, y, and z. Then the equation 



Ax + Ry + Cz + D = 



represents a plane, and the values of the constants A, B, C, and 

 D can be given in terms of the intercepts the plane makes on 

 the axes of reference. For if the intercepts are a, b, and c on 

 the axes OX, OY, and OZ respectively 



Then when x = a, y = 0, and z= hence Aa + D = 

 when y = b, x = 0, and z = hence B& + D = 

 when z = c, x = 0, and y = hence Cc + D = 



Therefore a = - -r-, b - ^, and c = -~ 

 A r> U 



ABC 

 But rf + D^ + D* = 



it' T* ~- x 



ABC 



Then -.-{-. -1 



a 6 c 



a? v z 



or _ + ^ + _=l 



a b c 



This gives the equation of a plane in terms of the inter- 

 cepts. 



Let OP (Fig. 23) be the perpendicular drawn from the origin 



*7* 77 2> 



to the plane whose equation is - + 7+- = !. 



a b c 



Let ojj, y lt z 1 be the co-ordinates of P ; and if a, (3, and y are 

 the angles made by OP with the axes OX, OY, and OZ respec- 

 tively, and I, m, n are the direction cosines of OP 



Then I = cos a = 



P 



m = cos B = 



n = cos V = 

 P 

 where p is the length of OP. 



Since OP is perpendicular to the plane, it is perpendicular to 

 the lines OA, OB, and OC in that plane, and therefore the tri- 

 angles POA, POB, and POC are right angled. 



