|Class 12 Maths Three Dimensional Geometry||Coplanarity of Two lines:Vector|
Coplanarity of Two lines:Vector
Consider 2 lines such that: -First line be represented as: - r->=a1 ⃗+λ b1⃗, it passes through a point A, having position vector a1⃗as and it is parallel to b1⃗.
Second line r-> = a2⃗+µ b2⃗, it passes through a point B,having position vector a2⃗and is parallel to b2⃗.
Thus AB ⃗= (a2⃗- a1⃗). The given lines are coplanar iff AB ⃗is perpendicular to (b1⃗x b2⃗).
Let the coordinates of A =(x1, y1, z1) and of B= (x2, y2, z2).
Direction ratios of b1⃗= (a1, b1, c1) and of b2⃗ =(a2,b2,c2).
AB ⃗=(x2-x1) î +(y2-y1) ĵ +(z2-z1) k̂
b1⃗= (a1 î+b1 ĵ+c1 k̂) and b2⃗= (a2 î,b2 ĵ,c2 k̂).
The lines will be coplanar iff AB ⃗. (b1⃗x b2⃗) =0.
Therefore in the Cartesian form ,it can be given as:-