If $\overline{a}=\hat{i}-\hat{k}$,$\overline{b}=x \hat{i}+\hat{j}+(1-x) \hat{k}$ and $\overline{c}=y \hat{i}+x \hat{j}+(1+x-y) \hat{k}$,then $\overline{a} \cdot(\overline{b} \times \overline{c})$ depends on

The volume of the parallelepiped having vertices at $O \equiv (0,0,0)$, $A \equiv (2,-2,1)$, $B \equiv (5,-4,4)$, and $C \equiv (1,-2,4)$ is:

If the volume of a tetrahedron whose conterminous edges are $\overline{a}+\overline{b}, \overline{b}+\overline{c}, \overline{c}+\overline{a}$ is $24$ cubic units,then the volume of the parallelepiped whose coterminous edges are $\overline{a}, \overline{b}, \overline{c}$ is

Let the volume of a parallelepiped whose coterminous edges are given by $\overrightarrow{u}=\hat{i}+\hat{j}+\lambda \hat{k}$,$\overrightarrow{v}=\hat{i}+\hat{j}+3 \hat{k}$ and $\overrightarrow{w}=2 \hat{i}+\hat{j}+\hat{k}$ be $1 \text{ cu. unit}$. If $\theta$ is the angle between the edges $\overrightarrow{u}$ and $\overrightarrow{w}$,then $\cos \theta$ can be

If $A \equiv (1, -1, 0)$,$B \equiv (0, 1, -1)$,and $C \equiv (-1, 0, 1)$,then the unit vector $\overline{d}$ such that $\overline{a}$ and $\overline{d}$ are perpendicular and $\overline{b}, \overline{c}, \overline{d}$ are coplanar is

If $\vec{r}$ is a vector perpendicular to both the vectors $2 \hat{i}+3 \hat{j}-4 \hat{k}$ and $3 \hat{i}-\hat{j}+\hat{k}$ and satisfies $\vec{r} \cdot(3 \hat{i}-3 \hat{j}+4 \hat{k})=5$, then $|\vec{r}|=$