Let a → = α i ^ + 3 j ^ - k ^ , b → = 3 i ^ - β j ^ + 4 k ^ and c → = i ^ + 2 j ^ - 2 k ^ where α , β ∈ R be three vectors. If the projection of a → on c → is 10 3 and b → × c → = - 6 i ^ + 10 j ^ + 7 k ^ , then the value of α + β equal to

Step-by-step solution

Given a → = α i ^ + 3 j ^ - k ^ , b → = 3 i ^ - β j ^ + 4 k ^ and c → = i ^ + 2 j ^ - 2 k ^ where α , β ∈ R be three vectors. Projection of a → on c → is a → · c → c → = 10 3 (given) ⇒ α + 6 + 2 1 + 4 + 4 = 10 3 ⇒ α = 2 and b → × c → = i ^ j ^ k ^ 3 - β 4 1 2 - 2 = - 6 i ^ + j ^ + k ^ ⇒ 2 β - 8 = - 6 ⇒ β = 1 Hence α + β = 3