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

Question

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, 15 17, 289 717, 289 2151, 17 2151

MARKS App · Accepted Answer

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 vector to the other plane is n 2 → = - 11 i ^ + 4 j ^ + 13 2 k ^ cos ⁡ θ = n 1 → · n 2 → n 1 → n 2 → = - 11 - 4 + 13 2 3 121 + 16 + 169 4 ⇒ cos ⁡ θ = - 17 3 717 ⇒ cos 2 ⁡ θ = 289 2151

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