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