If the vectors $a \hat{i}+\hat{j}+\hat{k}, \hat{i}+b \hat{j}+\hat{k}, \hat{i}+\hat{j}+c \hat{k}$ $(a \neq 1, b \neq 1, c \neq 1)$ are coplanar,then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is

Let $a=2 \hat{i}+\hat{j}-3 \hat{k}$ and $b=\hat{i}+3 \hat{j}+2 \hat{k}$. Then the volume of the parallelopiped having coterminous edges as $a, b$ and $c$,where $c$ is the vector perpendicular to the plane of $a, b$ and $|c|=2$ is

The volume (in cubic units) of the tetrahedron with edges $\hat{i}+\hat{j}+\hat{k}$,$\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}-\hat{k}$ is

If $\bar{a}=3 \hat{\imath}+\hat{\jmath}-\hat{k}, \bar{b}=2 \hat{\imath}-\hat{\jmath}+7 \hat{k}$ and $\bar{c}=7 \hat{\imath}-\hat{\jmath}+23 \hat{k}$ are three vectors,then which of the following statements is true?

Let $\vec{p} = 2\hat{i} + 3\hat{j} + a\hat{k}$,$\vec{q} = b\hat{i} + 5\hat{j} - \hat{k}$,and $\vec{r} = \hat{i} + \hat{j} + 3\hat{k}$. If $\vec{p}, \vec{q}, \vec{r}$ are coplanar and $\vec{p} \cdot \vec{q} = 20$,then the ordered pair $(a, b)$ is:

If $\bar{V} = 2\bar{i} + \bar{j} - \bar{k}$ and $\bar{W} = \bar{i} + 3\bar{k}$,and if $\bar{U}$ is a unit vector,then the maximum value of $[\bar{U} \bar{V} \bar{W}]$ is ...