The vector a = αi + 2j + βk lies in the plane of the vectors b = i + j and c = j + k and bisects the angle between b and c . Then which one of the fol - Mathematics MCQ AIEEE 2026 | ExamTest.live

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The vector a = α \hat{i} + 2 \hat{j} + β \hat{k} lies in the plane of the vectors b = \hat{i} + \hat{j} and c = \hat{j} + \hat{k} and bisects the angle between b and c . Then which one of the following gives possible values of α and β ?

Solution & Step-by-step Explanation

A vector bisecting the angle between b and c is given by: a = λ (\hat{b} + \overset{c}{^}) where \hat{b} = \frac{i ^ + j ^}{2} and \overset{c}{^} = \frac{j ^ + k ^}{2} . a = λ (\frac{i ^ + 2 j ^ + k ^}{2}) = \frac{λ}{2} \hat{i} + \frac{2 λ}{2} \hat{j} + \frac{λ}{2} \hat{k} Comparing this with a = α \hat{i} + 2 \hat{j} + β \hat{k} : y -component: \frac{2 λ}{2} = 2 ⟹ λ = 2 . α = \frac{λ}{2} = \frac{2}{2} = 1 . β = \frac{λ}{2} = \frac{2}{2} = 1 .

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The vector a = α \hat{i} + 2 \hat{j} + β \hat{k} lies in the plane of the vectors b = \hat{i} + \hat{j} and c = \hat{j} + \hat{k} and bisects the angle between b and c . Then which one of the following gives possible values of α and β ?

α = 2, β = 2

α = 1, β = 2

α = 2, β = 1

α = 1, β = 1

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