Question

Find the equation of the plane passing through ( a , b , c ) and parallel to the plane

Answer 1

The given plane passes through the point ( a,b,c ) and is parallel to the plane r → ⋅( i ^ + j ^ + k ^ )=2.

The equation of a plane parallel to a given plane, r → ⋅( x i → +y j → +z k → )=c is given by, r → ⋅( x i → +y j → +z k → )=λ

So, the equation of a plane parallel to the given plane, r → ⋅( i ^ + j ^ + k ^ )=2 is given by,

r → ⋅( i ^ + j ^ + k ^ )=λ(1)

The position vector of a point ( x,y,z ) is given as r → =x i → +y j → +z k → .

So, the position vector of the given point ( a,b,c ) is given as r → =a i ^ +b j ^ +c k ^ .

Substitute value of r → in equation (1),

( a i ^ +b j ^ +c k ^ )⋅( i → + j → + k → )=λ ( a+b+c )=λ

Substitute value of λ in equation (1),

r → ⋅( i → + j → + k → )=( a+b+c )(2)

Thus, the equation of the required plane is r → ⋅( i → + j → + k → )=( a+b+c )

Substitute r → =x i → +y j → +z k → in equation (2),

( x i ^ +y j ^ +z k ^ )⋅( i → + j → + k → )=( a+b+c ) x+y+z=a+b+c

Thus the equation of the plane passing through ( a,b,c )and parallel to the plane r → ⋅( i → + j → + k → )=2 is x+y+z=a+b+c.