The volume of tetrahedron with co-terminus edges $\bar{a}, \bar{b}, \bar{c}$ is $\frac{64}{3}$ cubic units, then volume of parallelopiped considering co-terminus edges given by the vectors $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$ is _________ cubic units.

If $\theta$ is an obtuse angle between vectors $\bar{a}$ and $\overline{\mathrm{b}}$ such that $|\overline{\mathrm{a}}|=5,|\overline{\mathrm{~b}}|=3$ and $|\overline{\mathrm{a}} \times \overline{\mathrm{b}}|=5 \sqrt{5}$ then $\bar{a} \cdot \bar{b}=$

If the vectors $\overline{\mathrm{a}}=\mathrm{c}\left(\log _7 x\right) \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \quad$ and $\overline{\mathrm{b}}=\left(\log _\gamma x\right) \hat{\mathrm{i}}+3 \mathrm{c}\left(\log _\gamma x\right) \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$ make obtuse angle for any $x>0$, then c belongs to

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