A vector a =α i +2 j +β k (α, β ∈ R ) lies in the plane of the vectors, b = i + j and c = i - j +4 k. If a bisects the angle between b and c, then:

A vector $\displaystyle \overrightarrow{\mathbf{a}}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}(\alpha, \beta \in \mathbf{R})$ lies in the plane of the vectors, $\displaystyle \overrightarrow{\mathrm{b}}=\hat{i}+\hat{j}$ and $\displaystyle \vec{c}=\hat{i}-\hat{j}+4 \hat{k}$. If $\displaystyle \vec{a}$ bisects the angle between $\displaystyle \overrightarrow{\mathrm{b}}$ and $\displaystyle \overrightarrow{\mathrm{c}}$, then :