Let 2 planes are being contained by the vectors α i ^ + 3 j ^ - k ^ ,   i ^ + α - 1 j ^ + 2 k ^ and 3 i ^ + 5 j ^ + 2 k ^ . If the angle between these 2 planes is θ , then the value of cos 2 ⁡ θ is equal to

Step-by-step solution

All given vectors are coplaner α 3 - 1 1 α - 1 2 3 5 2 = 0 ⇒ α 2 α - 2 - 10 - 3 2 - 6 - 1 5 - 3 α + 3 = 0 ⇒ 2 α 2 - 12 α + 12 - 8 + 3 α = 0 ⇒ 2 α 2 - 9 α + 4 = 0 ⇒ α = 4 , 1 2 A normal vector to the plane is i ^ j ^ k ^ 1 α - 1 2 3 5 2 = i ^ 2 α - 12 - j ^ - 4 + k ^ 8 - 3 α If α = 4 , then a normal vector to one of the plane is n 1 → = - 4 i ^ + 4 j ^ - 4 k ^ = - 4 i ^ - j ^ + k ^ If α = 1 2 , then a normal