THE POSITION OF A POINT IN SPACE 63 



and is therefore measured along OX ; it is negative when P lies 

 behind the plane ZOY, and is therefore measured along OXj. 



The plane ZOX is taken as a side vertical plane, and any line 

 drawn parallel to the axis OY will be perpendicular to that plane. 

 The y co-ordinate of a point P is positive when P lies to the right 

 of the plane ZOX, and is therefore measured along OY. It is 

 negative when P lies to the left of the plane ZOX, and is therefore 

 measured along OYj. The plane XOY is taken as a horizontal 

 plane, and any line drawn parallel to the axis OZ will be per- 

 pendicular to that plane. The z co-ordinate of a point P is 

 positive if P lies above the plane XOY, and is therefore measured 



FIG. 20. 



along OZ. It is negative when P lies below the plane XOY, and 

 is therefore measured along OZ X . 



Fig. 19 is drawn to illustrate this, the co-ordinates of Pj being 

 (I, 3, 2), while the co-ordinates of P 2 are ( 1, 3, 2). 

 42. Let P (Fig. 20) be a point whose co-ordinates are (x, y, z). 

 Then OA = as, OB = y, and OC = z. 



Since OC is perpendicular to the plane containing PC, the 

 angle PCO is a right angle, 



and OP 2 - OC 2 + PC 2 

 CP = OQ 

 OQ 2 = OB 2 + BQ 2 

 - OB 2 + OA 2 

 Hence OP 2 = OC 2 + OB 2 + OA 2 



But 

 and 



or OP = Vx 2 +y*+z 2 



