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 $\beta$ ?

JEE Main · 2008 · 27 Apr 2008 · Offline · Vector Algebra

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

Answer: (D)

A vector that bisects the angle between

b

and

c

points along the sum of their unit vectors. Both

b = \hat{i} + \hat{j}

and

c = \hat{j} + \hat{k}

have length

2

, so the bisector direction is proportional to

(\hat{i} + \hat{j}) + (\hat{j} + \hat{k}) = \hat{i} + 2 \hat{j} + \hat{k}

. Matching

a = α \hat{i} + 2 \hat{j} + β \hat{k}

to this direction, the middle component fixes the scale and then

α = 1

and

β = 1

JEE Main · 2008 · 27 Apr 2008 · Offline · Single correct option