Find the vector and Cartesian equations of a plane containing the two lines and Also show that the lines lies in the plane.

Given : Equations of lines -




To Prove : are coplanar.


To Find : Equation of plane.


Formulae :


1) Cross Product :


If are two vectors




then,



2) Dot Product :


If are two vectors




then,



3) Coplanarity of two lines :


If two lines are coplanar then



4) Equation of plane :


If two lines are coplanar then equation of the plane containing them is



Where,



Answer :


Given equations of lines are




Let,


Where,






Now,





Therefore,



= 40 + 10 + 24


= 74


……… eq(1)


And



= 60 + 30 – 16


= 74


……… eq(2)


From eq(1) and eq(2)



Hence lines are coplanar.


Equation of plane containing lines is



Now,



From eq(1)


4


Therefore, equation of required plane is





This vector equation of plane.


As



= 20x + 10y – 8z


Therefore, equation of plane is


20x + 10y – 8z = 74


20x + 10y – 8z – 74 = 0


10x + 5y – 4z – 37 = 0


This Cartesian equation of plane.


1