Find the vector equation of the line that passes through the point (2,-1,7) and is parallel to the line of intersection of the planes x + 2y - 3z = -6 and 3x - y + 2z = 4

Answer 1

We need the direction of their line of intersection....
double the 2nd plane equation, then add that to the 1st

x+2y-3z = -6
6x - 2y + 4z = 8
7x + z = 2
z = 2-7x

let x=1
then z = -5
back in the 1st
1 + 2y + 15 = -6
y = -11 -----------> point (1,-11,-5)

let x = 0
then z = 2
back in 1st
0+2y -6 = -6
y = 0 ------------> point (0,0,2)

direction vector of line of intersection of the planes = (1, -11, -7)

so one such vector equation is
r = (2,-1,7) + t(1,-11,-7)

(the question should have said, "Find a vector equation...."
since the answer I obtained is not unique)

Answer 2

We need the direction of their line of intersection? That's all the question mentioned... so is that still the correct answer? :/