Let a → = i ^ - α j ^ + β k ^ , b → = 3 i ^ + β j ^ - α k ^ and c → = - α i ^ - 2 j ^ + k ^ , where α and β are integers. If a → · b → = - 1 and b → · c → = 10 , then a → × b → · c → is equal to ______.

Step-by-step solution

We have, a → = 1 , - α , β , b → = 3 , β , - α and c → = - α , - 2 , 1 ; α , β ∈ I a → · b → = - 1 ⇒ 3 - α β - α β = - 1 ⇒ α β = 2       . . . 1 Now, b → · c → = 10 ⇒ - 3 α - 2 β - α = 10 ⇒ 2 α + β + 5 = 0       . . . 2 From equations 1 and 2 , ⇒ 2 α + 2 α + 5 = 0 ⇒ 2 α 2 + 2 + 5 α = 0