Let a → = i ^ + α j ^ + β k ^ , α , β ∈ R . Let a vector b → be such that the angle between a → and b → is π 4 and b → 2 = 6 , If a → · b → = 3 2 , then the value of α 2 + β 2 | a → × b → | 2 is equal to

Step-by-step solution

Given: b → 2 = 6 , a → · b → = 3 2 , a → = i ^ + α j ^ + β k ^ We know that, a → . b → = a → b → cos θ ⇒ 3 2 = 1 2 + α 2 + β 2 6 cos π 4 ⇒ 3 2 = 1 2 + α 2 + β 2 3 ⇒ 1 + α 2 + β 2 = 6 ⇒ 1 + α 2 + β 2 = 6 ⇒ α 2 + β 2 = 5 Also, a → × b → = a → b → sin θ ⇒ a → × b → = 6 6 sin π 4 = 6 2 ⇒ a → × b → 2 = 18 ⇒ α 2 + β 2 a → × b → 2 = 5 × 18 ⇒ α 2 + β 2 a → × b → 2 = 90