THE ANGLE BETWEEN TWO STRAIGHT LINES 67 



Now cos 1 

 1 



{(X 3 - xj* 



, 



2 + (2 3 - zj* + (x t - xtf 

 - 2 i) 2 - (* - **) 2 - (2/3 - !/2) 2 



Z 3 Z 2 ) } 



*-* 



*-* 



If a 3 , P 3 , and Y 3 are the angles which P^ makes with the axes 

 OX, OY, and OZ respectively, and 1 3 , m 3 , and n 3 are the corre- 

 sponding direction cosines, 



then 1 3 = cos a 3 = 



n 3 = cos y 3 = 



j. v 



Also, if a 2 , (3 2 , and y 2 are the angles which PiP 3 makes with 

 the axes OX, OY, and OZ respectively, and 1 2 , w 2 , and rc 2 are the 

 corresponding direction cosines, 



then 



= cos a 2 



m 



cos 3 = 



n = cos Y* = 



Hence cos X = 1 3 1 2 + 



Similarly it can be proved that 



cos 2 = 



Pz 



Pz 



n 3 n 2 



and 



+ 



+ n z n 1 



cos 3 = 



thereby giving the cosine of an angle between two lines in terms 

 of the direction cosines of those lines. 



In general, if is the angle between two lines whose direction 

 cosines are l v m^ n t and 1 2 , m 2 , n 2 respectively, 



then cos = /j/ 2 + m^m^ + n^n^ 



If the two lines are at right angles, 



then /!/ a + w^w, + w x n 2 = 



