how 2 proove 2 lines are coplaner?

Question

eg take lines (x-1)/2=(y-3)/4=z/-1
& (x-4)/3=(y-1)/-2=z have 2 b prooved coplaner.

show mathematical way of solving,if sum formula available

Answer 1

ok u can easily do it...
suppose if the lines are (x-x1)/L1 = (y-y1)/M1 = (z-z1)/N1 and
(x-x2)/L2 = (y-y2)/M2 = (z-z2)/N2

then the condition for coplanarity is :
Determinant of :

| x2-x1 y2-y1 z2-z1 |
| L1 M1 N1 |
| L2 M2 N2 |

= 0.
if this satisfies then lines are coplanar.For ur lines ,the determinant is:
| 3 -2 0 |
| 2 4 -1 |
| 3 -2 1 |
= 3(4-2) +2(2+3) + 0
=3*2 + 2*5
=16
Therefore these lines aren't coplanar.

And if the lines are in vector form like : r1 = a1 + k*b1 and
r2 = a2 + k*b2
then the condition for coplanarity is:
[ a1 b1 b2 ] = [a2 b1 b2] ,where [... ] is the scalar triple product.

Answer 2

If the lines are not parallel and coplaner, they have to intersect.