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 v

Question

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

TestHub · Accepted Answer

∵ 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 α + 3 α + 6 β + 3 β = 18 c → = α a → + β b → , α + β = 2 …(i) Since a → , b → and c → are co-planar hence a → × b → . c → = 0 Minimum value of c → - a → × b → . c → = Minimum of c → . c → = Minimum of c → 2 ∵ a → × b → . c → = 0 now c → = α a → + β b → and α + β = 2 c → = α 2 i + j - k + 2 - α i + 2 j + k = c → = 2 + α i + 4 - α j + k 2 - 2 α c → 2 = 6 α 2 - 2 α + 4 M i n i m u m c → 2 = 18 ∴ M i n i m u m o f a n 2 + b n + c = - D 4 a = - b 2 - 4 a c 4 a

Mathematics - Vector Question with Solution | TestHub

Doubts & Discussion