If the origin $O(0,0,0)$ and the points $P(2,3,4)$,$Q(1,2,3)$,and $R(x, y, z)$ are co-planar,then:

If $\vec{\alpha}$ is a unit vector, $\vec{\beta}=\hat{i}+\hat{j}-\hat{k}$, and $\vec{\gamma}=\hat{i}+\hat{k}$, then the maximum value of $[\vec{\alpha} \vec{\beta} \vec{\gamma}]$ is

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 the points $A(3,2,1)$,$B(4, x, 5)$,$C(4,2,-2)$,and $D(6,5,-1)$ are coplanar,then $x$ has the value:

If the four points $A(6,2,4)$,$B(1,3,5)$,$C(1,-2,3)$,and $D(6, k, 2)$ are coplanar,then $k=$

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