For λ 0 , let θ be the angle between the vectors a = i ^ + λ j ^ − 3 k ^ and b = 3 i ^ − j ^ + 2 k ^ . If the vectors a + b and a − b are mutually perpendicular, then the value of (14 cos θ ) 2

Question

For λ > 0 , let θ be the angle between the vectors a = i ^ + λ j ^ ​ − 3 k ^ and b = 3 i ^ − j ^ ​ + 2 k ^ . If the vectors a + b and a − b are mutually perpendicular, then the value of (14 cos θ ) 2 is equal to

TestHub · Accepted Answer

​ ( a + b ) ⋅ ( a − b ) = 0 , λ > 0 ∣ a ∣ 2 − ∣ b ∣ 2 = 0 → 1 + λ 2 + 9 = 9 + 1 + 4 ∴ λ = 2 , cos θ = ∣ a ∣ ⋅ ∣ b ∣ a − b ​ = 14 ​ ⋅ 14 ​ 3 − λ − 6 ​ 14 cos θ = 3 − 8 = − 5 ∴ ( 14 cos θ ) 2 = 25 ​

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