Volume of a parallelepiped with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is $12$ cubic units. The volume of a tetrahedron with coterminous edges $\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{a} + \vec{b} - \vec{c}$ will be ............. cubic units.

The value of $x$ for which the volume of the parallelepiped formed by the vectors $\vec{a} = \hat{i} + x\hat{j} + \hat{k}$, $\vec{b} = \hat{j} + x\hat{k}$, and $\vec{c} = x\hat{i} + \hat{k}$ is minimum, is

If $\bar{p}, \bar{q}$ and $\bar{r}$ are non-zero,non-coplanar vectors,then $[\bar{p}+\bar{q}-\bar{r} \quad \bar{p}-\bar{q} \quad \bar{q}-\bar{r}] = \_\_\_\_$

The volume of the parallelepiped whose edges are represented by $-12i + \alpha k$,$3j - k$,and $2i + j - 15k$ is $546$. Then $\alpha = $

Let the volume of a parallelepiped whose coterminous edges are given by $\overrightarrow{u}=\hat{i}+\hat{j}+\lambda \hat{k}$,$\overrightarrow{v}=\hat{i}+\hat{j}+3 \hat{k}$ and $\overrightarrow{w}=2 \hat{i}+\hat{j}+\hat{k}$ be $1 \text{ cu. unit}$. If $\theta$ is the angle between the edges $\overrightarrow{u}$ and $\overrightarrow{w}$,then $\cos \theta$ can be

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac{\pi}{3}$. If $\vec{a} \times \vec{b} + \vec{b} \times \vec{c} = p \vec{a} + q \vec{b} + r \vec{c}$,where $p, q$ and $r$ are scalars,then the value of $\frac{p^2 + 2q^2 + r^2}{q^2}$ is