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

$α = 1, β = 1$

It is given that $\vec a$ bisects the angle between $\vec b$ and $\vec c$. Therefore, $\vec a$ is parallel $\frac{\hat b+\hat c}{2}$.

So, $\vec a=λ(\hat b+\hat c)$

$⇒α\hat i+2\hat j+β\hat k=λ\left(\frac{\hat i+\hat j}{\sqrt{2}}+\frac{\hat j+\hat k}{\sqrt{2}}\right)$

$⇒α\hat i+2\hat j+β\hat k=\frac{λ}{\sqrt{2}}(\hat i+2\hat j+\hat k)$

$⇒α=\frac{λ}{\sqrt{2}},2=\sqrt{2}λ$ and $β=\frac{λ}{\sqrt{2}}⇒α=1,β=1$