The altitude through vertex $A$ of $\triangle A B C$ with position vectors of points $A, B, C$ as $\bar{a}, \bar{b}, \bar{c}$ respectively is

If $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ are unit vectors and $|\overline{\mathrm{a}}|=7$, $\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+\overline{\mathrm{b}} \times(\overline{\mathrm{c}} \times \overline{\mathrm{a}})=\frac{1}{2} \overline{\mathrm{a}}$, then angle between the vectors $\bar{a}$ and $\bar{c}$ and angle between the vectors $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ are respectively

Let $\bar{a}=\hat{i}+\hat{j}-\hat{k}$ and $\bar{c}=5 \hat{i}-3 \hat{j}+2 \hat{k}$ and if $\overline{\mathrm{b}} \times \overline{\mathrm{c}}=\overline{\mathrm{a}}$ then $|\overline{\mathrm{b}}|=$

If $\bar{a}=\hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{j}-\hat{k}$ then a vector $\bar{c}$ such that $\overline{\mathrm{a}} \times \overline{\mathrm{c}}=\overline{\mathrm{b}}$ and $\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=3$ is