Let a → = α i ^ + 2 j ^ - k ^ and b → = - 2 i ^ + α j ^ + k ^ , where α ∈ R . If the area of the parallelogram whose adjacent sides are represented by the vectors a → and b → is 15 α 2 + 4 , then the value of 2 a → 2 + a → · b → b → 2 is equal to

Step-by-step solution

Given, a → = α i ^ + 2 j ^ - k ^ , b → = - 2 i ^ + α j ^ + k ^ , Area of parallelogram = a ^ × b ^ a ^ × b ^ = α + 2 2 + α - 2 2 + α 2 + 4 2 Given a ^ × b ^ = 15 α 2 + 4 So, 2 α 2 + 4 + α 2 + 4 2 = 15 α 2 + 4 ⇒ α 2 + 4 2 = 13 α 2 + 4 ⇒ α 2 + 4 = 13     ∴   α 2 = 9 Now, 2 a → 2 + a → · b → b → 2 a → 2 = α 2 + 4 + 1 = α 2 + 5 | b → | 2 = 4 + &#