Determine the equation for a line passing through (2,-3,-7) and parallel to the line of intersection of the planes x+y-z=0 and 5x-6y+6z=5

Question

Please include steps and answers

Answer 1

By linear combinations, the two equations can be rearranged as:

 x = 5/11  and   z – y  = 5//11

 Thus, the unit vector perpendicular to one plane is (1 , 0, 0)

 and the unit vector perpendicular to the other plane is ( 0, -.707 ,  .707)

     I am using .707 in place of 1/sqrt(2)    --- easier to type

The cross product of these two unit vectors will be a vector parallel to the line of intersection of the planes.

 The cross product is ( 0 , -.707 , -.707 )

From this, a parametric expression for the desired line is

  (2, -3 - .707 k , -7  -.707 k)     for all real k.  

By eliminating k between the y and z components, an alternate form for the line is the two equations

  x = 2       z = y -4