A vector a → = α i ^ + 2 j ^ + β k ^ α , β ∈ R lies in the plane of the vectors, b → = i ^ + j ^ and c → = i ^ - j ^ + 4 k ^ . If a → bisects the angle between b → and c → , then

Question

A vector a → = α i ^ + 2 j ^ + β k ^ α , β ∈ R lies in the plane of the vectors, b → = i ^ + j ^ and c → = i ^ - j ^ + 4 k ^ . If a → bisects the angle between b → and c → , then, a → ⋅ i ^ + 3 = 0, a → ⋅ i ^ + 1 = 0, a → ⋅ k ^ + 2 = 0, a → ⋅ k ^ + 4 = 0

MARKS App · Accepted Answer

As, a → lies in the plane of the vectors b → and c → , So, a → = λ i ^ + j ^ 2 + i ^ + j ^ + 4 k ^ 3 2 = λ 3 2 3 i ^ + 3 j ^ + i ^ - j ^ + 4 k ^ = λ 3 2 4 i ^ + 2 j ^ + 4 k ^ Compare with a → = a i ^ + 2 j ^ + β k ^ 2 λ 3 2 = 2 ⇒ λ = 3 2 a → = 4 i ^ + 2 j ^ + 4 k ^ Not in option So, now consider a → = μ i ^ + j ^ 2 - i ^ - j ^ + 4 k ^ 3 2 a → = μ 3 2 3 i ^ + 3 j ^ - i ^ + j ^ - 4 k ^ = μ 3 2 2 i ^ + 4 j ^ - 4 k ^ Compare with a → = α i ^ + 2 j ^ + β k ^ 4 μ 3 2 = 2 ⇒ μ = 3 2 2 a → = i ^ + 2 j ^ - 2 k ^ a → . k ^ + 2 = 0

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