A nice way to find the equation of a plane in R3 is:
v1 = (d,e,f) - direction of one of the lines
v2 = (u,v,t) - direction of the other line
p = (a,b,c) - One known point
Calculate this determinant equaling 0
| (x - a) u d |
| (y - b) v e | = 0
| (z - c) t f |...