Let a , b , c ∈ R be such that a 2 + b 2 + c 2 = 1 . If a cos θ = b cos θ + 2 π 3 = c cos θ + 4 π 3 ,where θ = π 9 , then the angle between the vectors a i ^ + b j ^ + c k ^ and b i ^ + c j ^ + a k ^ is:

Step-by-step solution

Let a cos θ = b cos   θ + 2 π 3 = c cos   θ + 4 π 3 = k ⇒ a = k cos θ ,   b = k cos θ + 2 π 3 ,   c = k cos θ + 4 π 3 ⇒ a b + b c + c a = k 2 cos θ + 4 π 3 + cos θ + cos θ + 2 π 3 cos θ + 4 π 3 cos θ cos θ + 2 π 3 = k 2 cos θ + 2 cos θ + π . cos π 3 cos θ . cos θ + 2 π 3 . cos θ + 4 π 3 = k 2 cos θ - 2 cos θ . 1 2 cos θ . cos