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

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 $a, b,$ and $c$ are unit coplanar vectors,then the scalar triple product $[a, b, c]$ is equal to:

The value of $\alpha$,so that the volume of the parallelepiped formed by $\hat{i}+\alpha \hat{j}+\hat{k}$,$\hat{j}+\alpha \hat{k}$,and $\alpha \hat{i}+\hat{k}$ becomes minimum,is

If $a, b, c$ are any three non-coplanar vectors,then $[a + b, b + c, c + a] = $

For non-zero vectors $\vec{a}, \vec{b}, \vec{c}$,the condition $|(\vec{a} \times \vec{b}) \cdot \vec{c}| = |\vec{a}||\vec{b}||\vec{c}|$ holds if and only if: