Let $\vec{a} = \hat{i} - \hat{k}$,$\vec{b} = x\hat{i} + \hat{j} + (1 - x)\hat{k}$,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 for vectors $\bar{a}, \bar{b},$ and $\bar{c},$ $[\bar{a} \bar{b} \bar{c}] = 4,$ then $[\bar{a} \times \bar{b}, \bar{b} \times \bar{c}, \bar{c} \times \bar{a}] = \dots$

Let the volume of the tetrahedron with vertices $\hat{i}-\hat{j}-2\hat{k}$,$-2\hat{i}+\hat{j}-2\hat{k}$,$-\hat{i}-2\hat{j}+\hat{k}$,and $2\hat{i}+2\hat{j}+a\hat{k}$ be $\frac{20}{3}$. Then the integral value of $a$ is

The volume of a tetrahedron with coterminus edges $\bar{a}, \bar{b}, \bar{c}$ is $\frac{64}{3}$ cubic units. Then,the volume of a parallelepiped with coterminus edges given by the vectors $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$ is ... cubic units.

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{b}$,and $\vec{c} = \hat{j} - \hat{k}$ be three vectors such that $\vec{a} \times \vec{b} = \vec{c}$ and $\vec{a} \cdot \vec{b} = 1$. If the length of the projection vector of the vector $\vec{b}$ on the vector $\vec{a} \times \vec{c}$ is $l$,then the value of $3l^{2}$ is equal to $.....$

If the points $A(1,1,2), B(2,1, p), C(1,0,3)$ and $D(2,2,0)$ are coplanar,then the value of $p$ is