Let P be a plane passing through the points ( 1 , 0 , 1 ) , ( 1 , - 2 , 1 ) and ( 0 , 1 , - 2 ) . Let a vector a → = α i ^ + β j ^ + γ k ^ be such that a → is parallel to the plane P , perpendicular to i ^ + 2 j ^ + 3 k ^ and a → · ( i ^ + j ^ + 2 k ^ ) = 2 , then ( α - β + γ ) 2 equals______.

Step-by-step solution

Equation of plane is x - 1 y - 0 z - 1 1 - 1 2 1 - 1 1 - 0 0 - 1 1 + 2 = 0 ⇒ x - 1 y z - 1 0 2 0 1 - 1 3 = 0 ⇒ x - 1 6 - y 0 + z - 1 - 2 = 0 ⇒ 3 x - 1 - z + 1 = 0 ⇒ 3 x - z - 2 = 0 Since, a → = α i ^ + β j ^ + γ k ^ is parallel to 3 x - z - 2 = 0 , therefore normal vector of plane is perpendicular to a → , hence ⇒ 3 α - γ = 0     . . . 1 Now, a → ⊥ i ˙ + 2 j ^ + 3 k ^ ⇒ α + 2 β + 3 γ = 0