THE STRAIGHT LINE 65 



X 9 Xi 



Then I - cos a - 



z z, 

 n = cos Y = * , 



thus giving the direction cosines of any line in terms of the co- 

 ordinates of any two points taken on the line. It follows that if P 

 is any point on a line, the co-ordinates of P being (x, y, z) , and Q 

 is a given point on the same line, the co-ordinates of Q being 

 (a, b, c). 

 If r is the length of line between P and Q, 



, x a 



then I = cos a = 



r 



x a 

 or -, = r 



p 



it b 



m = cos B = 



r 



y-b 



or = r 



m 



z c 



n = cos v = 



r 



z c 



or = r 



n 



x- a y-b z-c 



Hence -, = = = r 



I m n 



This is known as the symmetrical equation to a straight line. 

 44. Let Pj, P 2 , and P 3 be three points whose co-ordinates are 



/<v ft i *y i ( ** I/ Bf I JJTl/i/T 1 II 9*1 



\ 1 9 / 1 > I/ > V 2' fj 2> 2/ ' *-*-"*-* V 1 -* *5> j '\> I/ * 



If P! be taken as the origin, the co-ordinates of P 2 will be 

 (x 2 x^), (j/ 2 2/i) and (z 2 z 1 ), while the co-ordinates of P 3 

 will be (# 3 Xj), (y 3 y^), (z 3 Zj). 



Hence (PjP^ 2 = p = (x 2 x^ 2 + (y 2 t^) 2 + (z 2 zj 2 



If P 2 be taken as the origin, the co-ordinates of P 3 will be 

 (*3 - a), (2/3 - 2/2). and (z 3 - z 2 ). 



Hence (P 2 P 3 ) 2 = p[ = (x 9 - x 2 ) 2 + (t/ 3 - t/ 2 ) 2 + (z 3 - z 2 ) 2 



Thus the three sides of the triangle PjPjPg can be determined 

 if the co-ordinates of the three angular points are known. 



E 



