64 PRACTICAL MATHEMATICS 



This gives the distance of a point P from the origin in terms 

 of the co-ordinates of that point. 



Let a, (3, y be the angles which OP makes with the axes OX, 

 OY, and OZ respectively. 



(1) The triangle POA is such that the base angle POA = a, 



/\ 

 and PAO = 90 since OA is perpendicular to AP. 



OA 



Hence cos a = -- 



Vx 2 + y z + z 2 



(2) The triangle POB is such that the base angle POB = 



/\ 

 and PBO = 90 since OB is perpendicular to BP. 



Hence cos (3 = = 



y 



(3) The triangle POC is such that the base angle POC = y, and 

 /\ 

 PCO = 90 since OC is perpendicular to CP. 



OC 



Hence cos y = T= 



The cosines of the angles a, (3, and y, which the line joining a 

 point to the origin makes with the axes OX, OY, and OZ respec- 

 tively, are spoken of as the " direction cosines " of that line, 

 and are usually denoted by I, m, and n. 



The sum of the squares of the three direction cosines is 1 ; for 



I 2 + m 2 + n 2 = _., , * , .., + .., . y ., , ., + 



+ y* + & or + y + z* x* + y* + & 

 = 1 



43. Let P! and P 2 be two points whose co-ordinates are 

 (xi, y\> z i) an d (x 2 , 2/2 z 2 ). If the axes of reference are so chosen 

 that P! is taken as the origin, then the co-ordinates of P 2 with 

 reference to these axes will be (x 2 x-^, (y z y^, and (z 2 zj. 



Then PjPg = V(x 2 - xj 2 + (y z - ytf + (z 2 - zj 2 , thus giving 

 the length of a line joining two points in terms of the co-ordinates 

 of those points. 



If a, (3, and y are the angles which the line PjP 2 makes with 

 the true axes of reference OX, OY, and OZ, they will also be the 

 angles which the line makes with the parallel axes PjX, PjY, 

 and PjZ. 



