54 



MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



2.602 The perpendicular from the point xi, y\, z\ upon the plane Ax + By 

 Cz + D = o is: 



2.603 6 is the angle between the two planes: 



+ Biy + Ciz + Di = o, 

 + B 2 y + C 2 z + D 2 = o, 



cos 



= 



C 2 2 



2.604 Equation of the plane passing through the three points fa, yi, Zi) (x 2} y 2} z 2 ) 



y\ 

 y* 



+ y 



Z 2 X? I 

 Z3 ^3 I 



>'2 I 



23 



THE RIGHT LINE 



2.620 The equations of a right line passing through the point x\, y\, Zi, and whose 

 direction cosines are I, m, n are: 



x - 

 / 



y - y\ _ z - 

 m n 



2.621 6 is the angle between the two lines whose direction cosines are /i, m\, n^ 

 and ^nhjfh: 



cos 6 = /i/ 2 + m i m 2 + n^n 2) 



sin 2 6 = (linh I 2 mi) 2 + (m\n 2 mtfi\f- + (n\l 2 n^\f. 



2.622 The direction cosines of the normal to the plane denned by the two lines 

 whose direction cosines are /i, mi, n\ and / 2 , m 2 n 2 are: 



sin ' sin0 ' sin 6 



2.623 The shortest distance between the two lines: 



is: 



- x 2 ) 



z- 



+ (yi - 



X - X 2 



k 



z - z 2 



- nJi) + (z\ - z 2 ] 



{ (min 2 mtfiif- + (nih nJiY + (/ i w 2 / 2 mi) 2 } J 

 2.624 The direction cosines of the shortest distance between the two lines 



are: 



