Determine two other (non-trivial) points on the line that passes through P(5, 6, -2) and is parallel to the line with equation Vector r = (2,-3,4) + t(1,1, -3), t∈R.

Answer 1

So it must have direction (1,1, -3) and also pass through the point (5,6,-2)

The equation for your new equation is
(x,y,z) = (5,6,-2) + s(1,1,-3)
to get points, let s be any real number
let s = 2, (x,y,z) = (5,6,-2) + (2,2,-6) = (7,8,-8)
let s = 1, (x,y,z) = (5,6,-2) + (1,1,-3) = (6, 7,-5)