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^{->}=a_{1} ⃗+λ b_{1}⃗, it passes through a point A, having position vector a_{1}⃗as and it is parallel to b_{1}⃗.

Second line r^{->} = a_{2}⃗+µ b_{2}⃗, it passes through a point B,having position vector a_{2}⃗and is parallel to b_{2}⃗.

Thus AB ⃗= (a_{2}⃗- a_{1}⃗). The given lines are coplanar iff AB ⃗is perpendicular to (b_{1}⃗x b_{2}⃗).

__Cartesian form:-__

Let the coordinates of A =(x_{1}, y_{1}, z_{1}) and of B= (x_{2}, y_{2}, z_{2}).

Direction ratios of b_{1}⃗= (a_{1}, b_{1}, c_{1}) and of b_{2}⃗ =(a_{2},b_{2},c_{2}).

AB ⃗=(x_{2}-x_{1}) î +(y_{2}-y_{1}) ĵ +(z_{2}-z_{1}) k̂

b_{1}⃗= (a_{1} î+b_{1} ĵ+c_{1} k̂) and b_{2}⃗= (a_{2} î,b_{2} ĵ,c_{2} k̂).

The lines will be coplanar iff **AB ****⃗****. (b _{1}**

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

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