68 PRACTICAL MATHEMATICS 



Example 1. The co-ordinates of two points P and Q are (3, 7, 5) 

 and (5, 2, 8) respectively. Find the length of PQ, its direction 

 cosines, and the angles it makes with the axes of reference. 



Taking P as the origin, the co-ordinates of Q are (2, 5, 3), 



and PQ = A/2 2 + ( - 5) 2 + 3 2 

 = 6-164 



= 0-3244 



a = 71 4' 



P = 606l 



= - 0-8112 

 P = 144 12' 



= 0-4866 

 y = 60 53' 



Example 2. The co-ordinates of three points P, Q, and R are 

 (3, 6, 2), '(5, 9, 7), and (8, 3, 9) respectively. Find the lengths of 

 the lines joining these points, the angles between the lines, and 

 the lengths of the perpendiculars drawn from each point to the 

 opposite line. 



Q 



6-164. 



Taking P as the origin, the co-ordinates of Q are (2, 3, 5), and 

 the co-ordinates of R are (5, 3, 7) . 



Then PQ = A/2 2 + 3 2 + 5 2 



= 6-164 



PR= A/5 2 + ( - 3) 2 + 7 2 

 = 9-110 



Taking Q as the origin, the co-ordinates of R are (3, 6, 2). 



QR= V3 2 + ( - 6) 2 + 2 2 



= 7 



