The volume of the tetrahedron whose coterminous edges are represented by

$$ \bar{a}=-12 \hat{i}+p \hat{k}, \bar{b}=3 \hat{j},-\hat{k}, \bar{c}=2 \hat{i}+\hat{j}-15 \hat{k} $$

570 cu. units, then $\mathrm{p}=$

The maximum value and minimum value of the volume of the parallelopiped having coterminous edges $\hat{\mathrm{i}}+x \hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{j}}+x \hat{\mathrm{k}}$ and $x \hat{\mathrm{i}}+\hat{\mathrm{k}}$ are respectively

Let $\overline{\mathrm{a}}$ and $\overline{\mathrm{c}}$ be unit vectors at an angle $\frac{\pi}{3}$ with each other. If $(\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})) \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{c}})=5$, then $[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]=$

Let $\bar{a}, \bar{b}$, and $\bar{c}$ be unit vectors. Suppose that $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=0$ and if the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ is $\frac{\pi}{6}$, then $\overline{\mathrm{a}}$ is