The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2 x + 3 y + 4 z = 5 which is perpendicular to the plane x - y + z = 0 is

Step-by-step solution

Planes passing through line of intersection of the planes x + y + z = 1 and 2 x + 3 y + 4 z = 5 is x + y + z - 1 + λ 2 x + 3 y + 4 z - 5 = 0 ⇒ ( 1 + 2 λ ) x + ( 1 + 3 λ ) y + ( 1 + 4 λ ) z - ( 1 + 5 λ ) = 0 Since required plane is perpendicular to x - y + z = 0 , hence 1 ⋅ ( 1 + 2 λ ) - 1 ⋅ ( 1 + 3 λ ) + 1 ⋅ ( 1 + 4 λ ) = 0 ⇒ λ = - 1 3 Hence, required plane is x - z + 2 = 0 i.e. r → ⋅ ( i ^ - k ^ ) + 2 = 0