You're on it so far. Keep going. For notation's sake, let the two vector equations represent lines: and . You correctly determined that the line sought will be of the form for some "starting vector" . Notice that the choice for the starting vector is not unique, it can be any point on the line . Since we know intersects , without losing generality, let for some . In other words, line "starts" at its intersection with and progresses in the direction towards . At some value of , will intersect line . We now have the ingredients to set up a system of equations: There exists integers such that . This yields a 3x4 matrix with variables that can be reduced to find a unique solution. I'll let you do the grunt work for yourself, but the answer, so you can check it, is . (By the way, what would it mean if said matrix were singular? )