862 ANALYTIC GEOMKTHY [('... III. 



therefore the points where the given line pierces the coordi- 

 nate planes are 



,* 0), =0, = _ ,S S =5, 0, - (4) 



223. Reduction of the general equations of a straight line 

 to a standard form. Determination of the direction angles 

 and traces. 



I. Third standard form: traces. The traces of a straight 

 line have the same equations as have the planes of }:<>]<(- 

 tion of the straight line upon the coordinate planes, respec- 

 tively. They may be obtained, therefore (Art. 210), ly 

 eliminating in turn each of the variables z, y, x from the 

 given equations. 



This may be illustrated by a numerical example. 



Given the equations 



-5 = 0, x + 2y-2z = 3, . . . (1) 



representing a straight line. Eliminating z, y, and x, successively, the 

 equations 



7x + 6y-13 = 0, 2* + 32 -2=0, 4y-7z-4 = . . . (J) 



are obtained, each representing a plane through the given line and ]- 

 pendicular to a coordinate plane. Therefore these equations are also the 

 equations of the traces of the line, in the xy-, zx-, and yz-planes, respectively. 



II. first standard form: direction angles . The method <!' 

 reducing the general equations of a straight line to the first 

 standard form, and finding its direction angles, can also be 

 illustrated by a numerical case. 



Considering still the line whose equations are (1) above, and whose 

 traces are given by equations (2); and taking the equations of any two 

 of its traces, e.g., 



2* + 3s -2 = 0, 4y-7z-4 = 0; . . . (3) 



