The volume of the parallelepiped whose conterminous edges are $\vec{a} = \hat{i} - \hat{j} + \hat{k}$,$\vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k}$,and $\vec{c} = 3\hat{i} - 5\hat{j} + 2\hat{k}$ is:

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 the points with position vectors $\hat{i}-\hat{j}+\hat{k}$, $2 \hat{i}-\hat{k}$, $\hat{j}+2 \hat{k}$ and $\hat{i}+\hat{j}+\lambda \hat{k}$ are coplanar, then the magnitude of the vector $6 \lambda \hat{i}-3 \hat{j}+6 \hat{k}$ is

Three lines are drawn from the origin $O$ with direction ratios proportional to $(1, -1, 1)$, $(2, -3, 0)$, and $(1, 0, 3)$. The three lines are

If the volume of a tetrahedron,whose vertices are with position vectors $\hat{i}-6 \hat{j}+10 \hat{k}$,$-\hat{i}-3 \hat{j}+7 \hat{k}$,$5 \hat{i}-\hat{j}+\lambda \hat{k}$ and $7 \hat{i}-4 \hat{j}+7 \hat{k}$ is $11$ cubic units,then the value of $\lambda$ is:

If the volume of the parallelepiped whose coterminous edges are along the vectors $\bar{a}, \bar{b}, \bar{c}$ is $12$,then the volume of the tetrahedron whose coterminous edges are $\bar{a}+\bar{b}, \bar{b}+\bar{c}$ and $\bar{c}+\bar{a}$ is