:l;4 .1 YALYTIC BXOMWTST [Cn. in. 



and therefore, by equation [ in], tin- angle between the t\\<> 

 lines is given by the equation 



.... [29] 



Again, the angle between the straight line 



a b c 



and the plane 



Ax + By + Cz + D= . . . (5) 



is the complement of the angle between the line (-\) and tin- 

 perpendicular to the plane (4) from the origin. Therefore, 

 by equations [10] and [21], and Art. 223, II, the required 

 angle is given by the equation 



""= <M+ft - +< *L.^ [30] 



Conditions for perpendicularity and parallelism precisely 

 like those of Art. 219 may be obtained from equations [-J'.'J 

 and [30]. 



EXAMPLES ON CHAPTER III 



1. Find the equations of a line through the points (1, 2, 3) and 

 (3, 2, 1). 



2. Find the equation of a plane through three points (1, 2, 3), 

 (3, 2, 1), and (2, 3, 1). 



3. Write the equations of the straight line through the point 

 (1, 2, 3), and having its direction cosines proportional to \/3, 1, L' \ '. 



4. What are the traces of the line of Ex. 1 upon the coordinate 

 planes? Where does the line pierce those planes? 



5. Find the equations of a straight line through the point (1, 2, 

 and perpendicular to the plane x + 2y + 3c = 6. 



Reduce to the intercept and normal forms, and determine which 

 octant each plane cuts : 



