Let θ be the angle between the planes P 1 = r → · i ^ + j ^ + 2 k ^ = 9 and P 2 = r → · 2 i ^ - j ^ + k ^ = 15 . Let L be the line that meets P 2 at the point 4 , - 2 , 5 and makes an angle θ with the normal of P 2 . If α is the angle between L and P 2 then tan 2 θ cot 2 α is equal to _____ .

Step-by-step solution

Since, θ be the angle between the planes P 1 : r → · i ^ + j ^ + 2 k ^ = 9 and P 2 : r → · 2 i ^ - j ^ + k ^ = 15 , so cos θ = i ^ + j ^ + 2 k ^ · 2 i ^ - j ^ + k ^ 6 ⇒ cos θ = 3 6 = 1 2 ⇒ θ = π 3 And, So, α = π 2 - π 3 = π 6 Hence tan 2 θ cot 2 α = 3 × 3 = 9