I am interested in finding a vector equation of intersection of the two planes $4x + y - 7z = 0$ and $2x - 3y + 4z = -8$. (This is from a past A-level Further Maths paper).
The mark scheme substitutes in $x=0$, $y = 0$ and $z = 0$ respectively to get the points $(0,56/17,8/17)$, $(-28/15,0,-16/15)$, $(-4/7,16/7,0)$ and then deduces that the direction vector of the line of intersection is $17{\bf i}+30{\bf j} + 14{\bf k}$. Then, the equation of the line is expressed in the form ${\bf r} = 56/17{\bf j} + 8/17{\bf k} + \lambda (17{\bf i}+30{\bf j} + 14{\bf k}).$
How is this direction vector deduced?