  Question

The vector $$\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k$$ lies in the plane of the vectors $$\overrightarrow b = \widehat i + \widehat j$$ and $$\overrightarrow c = \widehat j + \widehat k$$ and bisects the angle between $$\overrightarrow b$$ and $$\overrightarrow c .$$ Then, which one of the following gives possible values of $$\alpha$$ and $$\beta ?$$

A
α=2,β=2  B
α=1,β=2  C
α=2,β=1  D
α=1,β=1  Solution

The correct option is D $$\alpha = 1,\,\beta = 1$$$$\vec { a } =\lambda \left( \vec { b } +\vec { c } \right)$$$$(\because \vec { a }$$ lies in plane of $$\vec { b }$$ and $$\vec { c }$$ and bisects the angle between $$\vec { b }$$  and $$\vec { c } )$$$$\Rightarrow \quad \alpha \vec { i } +2\vec { j } +\beta \vec { k } =\lambda \left( \cfrac { \vec { i } +\vec { j } }{ \sqrt { 2 } } +\cfrac { \vec { j } +\vec { k } }{ \sqrt { 2 } } \right)$$$$\Rightarrow \quad \alpha \vec { i } +2\vec { j } +\beta \vec { k } =\lambda \left( \cfrac { \vec { i } +2\vec { j } +\vec { k } }{ \sqrt { 2 } } \right)$$$$\Rightarrow \quad \lambda =\sqrt { 2 } \alpha \quad ,\quad \lambda =\sqrt { 2 } \quad and\quad \lambda =\sqrt { 2 } \beta$$$$\therefore \quad \alpha =\beta =1$$Maths

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