Question

Find parametric equations for the line which is the intersection of the planes with equations

x+y+z=1

x-y+2z=0

Answer 1

the equations of two planes are

   x + y + z = 1            and    x - y + 2 z  =0 

perpendiculars to these planes are given by the vectors :

     ( 1, 1, 1 ) or i + j + k             (1, -1, 2 )  or  i - j + k 

cross product of these two vectors :
 
   - i X j + i X k + j X i + j X k + k X i - k X j  =  2 i   + 0 j  - 2 k  or (2, 0, -2)

   Let z = 0,  then solving  x + y +z = 1 and x - y +2 z = 0  gives,
           (1/2 , 1/2, 0) is a point on both planes.

    Equation of the line intersection of both planes and parallel to (2, 0 , -2) is:

                x = 1/2 + 2 t,         y = 1/2 + 0 t     z = 0 + -2 t
         or  x = 1/2 + 2 t   ,  y = 1/2  , z = -2 t