Let a → = 2 i ^ + j ^ - k ^ and b → = i ^ + 2 j ^ + k ^ be two vectors. Consider a vector c → = α a → + β b → ,   α ,   β ∈ R . If the projection of c → on the vector a → + b → is 3 2 , then the minimum value of c → - a → × b → . c → equals

Step-by-step solution

∵ a → + b → = 3 i + 3 j ∵ a → = 2 i + j - k , a → = 6 ⇒ a → + b → = 3 2 , b → = i + 2 j + k , b → = 6 a → . b → = 2 + 2 - 1 = 3 Now, as given projection of a → + b → on c → = 3 2 a → + b → a → + b → . c → = 3 2 a → + b → . α a → + β b → = 18 α a → 2 + α a → . b → + β b → 2 + β a → . b → = 18 as given 6