The vector $\bar{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}$ lies in the plane of the vectors $\overline{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$ and $\overline{\mathrm{c}}=\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and bisects the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$. Then which one of the following gives possible values of $\alpha$ and $\beta$ ?

A unit vector coplanar with $\hat{i}+\hat{j}+\hat{k}$ and $2 \hat{i}+\hat{j}+\hat{k}$ and perpendicular to $\hat{i}+\hat{j}-\hat{k}$ is

Let $\bar{a}=3 \hat{i}-\alpha \hat{j}+\hat{k}$ and $\bar{b}=\hat{i}+\alpha \hat{j}+3 \hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$, is $8 \sqrt{3}$ sq. units, then $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}$ is equal to

Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\overline{\mathrm{c}}=\hat{\mathrm{a}}+2 \hat{\mathrm{~b}}$ and $\overline{\mathrm{d}}=5 \hat{\mathrm{a}}+4 \hat{\mathrm{~b}}$ are perpendicular to each other, then the angle between $\hat{a}$ and $\hat{b}$ is