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 $\overrightarrow{u}, \overrightarrow{v}$ 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.

Let $a, b, c$ be three distinct real numbers,none equal to $1$. If the vectors $a \hat{i}+\hat{j}+\hat{k}$,$\hat{i}+b \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+ c \hat{k}$ are coplanar,then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is equal to

For what value of $a$ is the volume of the parallelepiped formed by the vectors $\hat{i} + a\hat{j} + \hat{k}$,$\hat{j} + a\hat{k}$,and $a\hat{i} + \hat{k}$ minimum?

If $\vec{a} = \hat{i} - \hat{k}$, $\vec{b} = x\hat{i} + \hat{j} + (1 - x)\hat{k}$ and $\vec{c} = y\hat{i} + x\hat{j} + (1 + x - y)\hat{k}$, then the scalar triple product $[\vec{a} \vec{b} \vec{c}]$ depends on

If the vectors $ai + j + k$,$i + bj + k$,and $i + j + ck$ $(a \ne 1, b \ne 1, c \ne 1)$ are coplanar,then the value of $\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} = $