Two lines have equations r_1=(1,-1,2)+a(-1,3,4) and r_2=(c,-4,0)+b(0,3,2). If the lines intersect find c:

If the lines intersect the position vectors r_1 and r_2 must be equal at the point of intersection, so: (1,-1,2)+a(-1,3,4)=(c,-4,0)+b(0,3,2) which gives three equations for the three components: 1-a=c, -1+3a=-4+3b, 2+4a=2b. From the last two obtain b=5 and a=2 then substitute in the first to find c=-1.