The volume of the tetrahedron formed by the coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is $3$. Then the volume of the parallelepiped formed by the coterminous edges $\vec{a} + \vec{b}, \vec{b} + \vec{c}, \vec{c} + \vec{a}$ is

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{b} = \hat{i} - \hat{j} + 2\hat{k}$,and $\vec{c} = x\hat{i} + (x-2)\hat{j} - \hat{k}$. If the vector $\vec{c}$ lies in the plane of $\vec{a}$ and $\vec{b}$,then $x$ equals:

If the vectors $\vec{b}, \vec{c}, \vec{d}$ are not coplanar,then the vector $(\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d})+(\vec{a} \times \vec{c}) \times(\vec{d} \times \vec{b})+(\vec{a} \times \vec{d}) \times(\vec{b} \times \vec{c})$ is

Let $\overrightarrow{u}, \overrightarrow{v}$ and $\overrightarrow{w}$ be vectors in three-dimensional space,where $\overrightarrow{u}$ and $\overrightarrow{v}$ are unit vectors which are not perpendicular to each other and $\overrightarrow{u} \cdot \overrightarrow{w}=1, \overrightarrow{v} \cdot \overrightarrow{w}=1, \overrightarrow{w} \cdot \overrightarrow{w}=4$. If the volume of the parallelepiped,whose adjacent sides are represented by the vectors $\overrightarrow{u}, \overrightarrow{v}$ and $\overrightarrow{w}$,is $\sqrt{2}$,then the value of $|3\vec{u}+5\vec{v}|$ is.

The volume of the parallelepiped formed by the vectors $\hat{i} + m \hat{j} + \hat{k}$,$\hat{j} + m \hat{k}$,and $m \hat{i} + \hat{k}$ becomes minimum when $m$ is

If the volume of the parallelepiped formed by three non-coplanar vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $4$ cubic units,then $[\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]$ is equal to