Question

how to prove in vectors three lines are coplaner

Answer 1

For three vectors a⃗ ,b⃗ ,c⃗ a→,b→,c→ the scalar triple product a⃗ ⋅(b⃗ ×c⃗ )a→⋅(b→×c→) is the (signed) volume of the parallelepiped defined by the three vectors (if you arranged them with a common base point in space).

This signed volume is zero precisely when the three vectors are coplanar.