Prove that the lines and are coplanar. Also find the equation of the plane containing these lines.

Given : Equations of lines –


Line 1 :


Line 2 :


To Prove : Line 1 & line 2 are coplanar.


To Find : Equation of plane.


Formulae :


1) Coplanarity of two lines :


If two lines are given by,


and


, then these lines are coplanar, if



2) Equation of plane :


The equation of plane containing two coplanar lines


& is given by,



Answer :


Given lines –


Line 1 :


Line 2 :


Here, x1 = 0 , y1 = 2 , z1 = -3 , a1 = 1 , b1 = 2 , c1 = 3


x2 = 2 , y2 = 6 , z2 = 3 , a2 = 2 , b2 = 3 , c2 = 4


Now,






= - 2 + 8 – 6


= 0



Hence, given two lines are coplanar.


Equation of plane passing through line1 and line 2 is given by,






-x + 2y – 4 – z – 3 = 0


- x + 2y – z – 7 = 0


x – 2y + z + 7 = 0


Therefore, equation of plane is


x – 2y + z + 7 = 0


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