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).

Cartesian form:-

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 . (b1x b2) =0.

Therefore in the Cartesian form ,it can be given as:-

 

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