Question
The vector $$\overrightarrow a = \alpha \hat i + 2\hat j + \beta \hat k$$ lies in the plane of the vectors $$\overrightarrow b = \hat i + \hat j$$ and $$\overrightarrow c = \hat j + \hat k$$ and bisects the angle between $$\overrightarrow b $$ and $$\overrightarrow c $$. Then which one of the following gives possible values of $$a$$ and $$b\,?$$
A.
$$\alpha = 2,\,\beta = 2$$
B.
$$\alpha = 1,\,\beta = 2$$
C.
$$\alpha = 2,\,\beta = 1$$
D.
$$\alpha = 1,\,\beta = 1$$
Answer :
$$\alpha = 1,\,\beta = 1$$
Solution :
$$\eqalign{
& \because \,\overrightarrow a {\text{ lies in the plane of }}\overrightarrow b {\text{ and }}\overrightarrow c \cr
& \therefore \,\overrightarrow a = \overrightarrow b + \lambda \overrightarrow c \cr
& \Rightarrow \alpha \hat i + 2\hat j + \beta \hat k = \hat i + \hat j + \lambda \left( {\hat j + \hat k} \right) \cr
& \Rightarrow \alpha = 1,\,2 = 1 + \lambda ,\,\beta = \lambda \cr
& \Rightarrow \alpha = 1,\,\,\beta = 1 \cr} $$