The set of all α , for which the vectors a = α t i ^ + 6 j ^ − 3 k ^ and b = t i ^ − 2 j ^ − 2 α t k ^ are inclined at an obtuse angle for all t ∈ R , is

Question

The set of all α , for which the vectors a = α t i ^ + 6 j ^ ​ − 3 k ^ and b = t i ^ − 2 j ^ ​ − 2 α t k ^ are inclined at an obtuse angle for all t ∈ R , is

TestHub · Accepted Answer

​ a = α i ^ + 6 j ^ ​ − 3 k ^ b = t i ^ − 2 j ^ ​ − 2 α k ^ so a b < 0 , ∀ t ∈ R α t 2 − 12 + 6 α t < 0 α t 2 + 6 α t − 12 < 0 , ∀ t ∈ R α < 0 , and D < 0 36 α 2 + 48 α < 0 12 α ( 3 α + 4 ) < 0 3 − 4 ​ < α < 0 ​ also for a = 0 , a ⋅ b < 0 hence a α ∈ ( 3 − 4 ​ , 0 ]

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