If a → = α i ^ + β j ^ + 3 k ^ , b → = - β i ^ - α j ^ - k ^ and c → = i ^ - 2 j ^ - k ^ such that a → · b → = 1 and b → · c → = - 3 , then 1 3 ( ( a → × b → ) · c → ) is equal to _______.

Question

If a → = α i ^ + β j ^ + 3 k ^ , b → = - β i ^ - α j ^ - k ^ and c → = i ^ - 2 j ^ - k ^ such that a → · b → = 1 and b → · c → = - 3 , then 1 3 ( ( a → × b → ) · c → ) is equal to _______.,

MARKS App · Accepted Answer

a → · b → = 1 ⇒ - α β - α β - 3 = 1 ⇒ - 2 α β = 4 ⇒ α β = - 2 … 1 b → · c → = - 3 ⇒ - β + 2 α + 1 = - 3 β - 2 α = 4 … 2 Solving 1 and 2 , ( α , β ) = ( - 1 , 2 ) 1 3 [ a → b → c → ] = 1 3 α β 3 - β - α - 1 1 - 2 - 1 = 1 3 - 1 2 3 - 2 1 - 1 1 - 2 - 1 R 1 → R 1 + R 3 = 1 3 0 0 2 - 2 1 - 1 1 - 2 - 1 = 1 3 2 4 - 1 = 2

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