Let a → and b → be two vectors such that a → = 1 , b → = 4 and a → ⋅ b → = 2 . If c → = 2 a → × b → − 3 b → and the angle between b → and c → is α , then 192 sin 2 α is equal to _________

Step-by-step solution

Given, a → and b → be two vectors such that a → = 1 , b → = 4 and a → ⋅ b → = 2 . Also, given c → = 2 a → × b → − 3 b → Taking dot product with b → both side we get, ⇒ b → ⋅ c → = 2 a → × b → ⋅ b → − 3 b 2 ⇒ b → c → cos α = 0 − 3 b → 2 ⇒ c → cos α = − 12 as b → = 4 Now, solving a → ⋅ b → = 2 ⇒ a → b → cos θ = 2 ⇒ cos θ = 1 2 ⇒ θ = π 3 Now, solving c → 2 = 2 a → × b → − 3 b → 2 ⇒ c → 2 = 64 × 3 4 + 144 = 192 Now, squaring both side of c → cos α = − 12 we get, c → 2 cos 2 α = 144 ⇒ 192 cos 2 α = 144 ⇒ 192 1 - sin 2 α