If $\bar{a}=4 \hat{i}+3 \hat{j}+\hat{k}, \bar{b}=\hat{i}-2 \hat{j}+2 \hat{k}$ then $\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times \overline{\mathrm{b}})))=$

If $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are unit vectors and $\theta$ is the angle between them, then $\overline{\mathrm{a}}+\overline{\mathrm{b}}$ is a unit vector when $\theta$ is

The magnitude of a vector which is orthogonal to the vector $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and is coplanar with the vectors $\hat{i}+\hat{j}+2 \hat{k}$ and $\hat{i}+2 \hat{j}+\hat{k}$ is

Let $\overline{\mathrm{OA}}=\overline{\mathrm{a}}, \overline{\mathrm{OB}}=\overline{\mathrm{b}}$ and if the vector along the angle bisector of $\angle \mathrm{AOB}$ is given by $x \frac{\overline{\mathrm{a}}}{|\overline{\mathrm{a}}|}+y \frac{\overline{\mathrm{~b}}}{|\overline{\mathrm{~b}}|}$ then