/..N.I /.)// UBOMBTR1 





If P 8 bisects the line 

 forms 



formulas [S] take the simpler 

 * 



206. Angle between two radii vectores. Angle between two 



lines. Let P l 



and 



, 



? a ) be 



INVU given points, aud the angle included by the radii 



tores pj and p r Then tin- jr>- 

 jections upon (?P X of the line 

 and of the broken line 

 are equal (Art. 17); 

 hence. 



But 



hence, 



cos 



cos cos 



cos 



cos 



cos 



cos 



COS 



COS ai COS a^ 4- COS Pi COS fo + COS fl COS Y* 



and this relation determines the required angle 6. 



It follows, since any t\vo straight lines in space have their 

 directions given by the direction angles of radii vectores 

 which are parallel to them, respectively, that formula [In] 

 applies as well to the angle 6 between any tNvo straight lines 

 in space, whose direction angles are given. 



Two special cases arise, that of parallel and that of perpen- 

 dicular lines. If the two given lines are parallel, evidently 



a = a 8 = 6 v = v ft 1 1 



