If $[\bar{a} \bar{b} \bar{c}] = 4$,then the volume (in cubic units) of the parallelepiped with $\bar{a} + 2 \bar{b}, \bar{b} + 2 \bar{c}$ and $\bar{c} + 2 \bar{a}$ as coterminal edges,is

If the position vectors of the vertices $A, B$ and $C$ are $6i$,$6j$ and $k$ respectively with respect to the origin $O$,then the volume of the tetrahedron $OABC$ is

If the vectors $4i+11j+mk$,$7i+2j+6k$,and $i+5j+4k$ are coplanar,then $m$ is

Let $\vec{c}$ be a vector coplanar with the unit vectors $\vec{a}$ and $\vec{b}$,and let $\vec{d}$ be the unit vector perpendicular to $\vec{a}$,$\vec{b}$,and $\vec{c}$. If $[\vec{a} \vec{b} \vec{d}] \vec{c} - [\vec{a} \vec{b} \vec{c}] \vec{d} = \hat{i} + 2\hat{j} + 2\hat{k}$ and the angle between $\vec{a}$ and $\vec{b}$ is $30^{\circ}$,then $|\vec{c}| =$

If three vectors $2\hat{i}-\hat{j}-\hat{k}$,$\hat{i}+2\hat{j}-3\hat{k}$ and $3\hat{i}+\lambda\hat{j}+5\hat{k}$ are coplanar,then the value of $\lambda$ is

If $a(\alpha \times \beta)+b(\beta \times \gamma)+c(\gamma \times \alpha)=0$ and at least one of the scalars $a, b, c$ is non-zero,then the vectors $\alpha, \beta, \gamma$ are