Let a → = a 1 i ^ + a 2 j ^ + a 3 k ^ , a i > 0 , i = 1 , 2 , 3 be a vector which makes equal angles with the coordinate axes O X , O Y and O Z . Also, let the projection of a → on the vector 3 i ^ + 4 j ^ be 7 . Let b → be a vector obtained by rotating a → with 90 ° . If a → , b → and x -axis are coplanar, then projection of a vector b → on 3 i ^ + 4 j ^ is equal to

Step-by-step solution

Given a → = a 1 i ^ + a 2 j ^ + a 3 k ^ Since the vector makes equal angles with the coordinate axes O X , O Y and O Z , so a → = λ 1 3 i ^ + 1 3 j ^ + 1 3 k ^ = λ 3 i ^ + j ^ + k ^ Now projection of a → on b → = 7 ⇒ λ 3 i ^ + j ^ + k ^ · 3 i ^ + 4 j ^ 5 = 7 ⇒ λ = 5 3 So a → = 5 i ^ ^ + j ^ + k ^ Let b → = 5 α i ^ + j ^ + k ^ + β i ^ Since both vectors are perpendicular a → · b → = 0 ⇒ 25 α 3 + 5 β =