If the vertices of a tetrahedron are $\vec{a} = \vec{j} + 2\vec{k}$,$\vec{b} = 3\vec{i} + \vec{k}$,$\vec{c} = 4\vec{i} + 3\vec{j} + 6\vec{k}$,and $\vec{d} = 2\vec{i} + 3\vec{j} + 2\vec{k}$,find its volume.

The volume of the parallelepiped formed by the vectors $\hat{i} + m \hat{j} + \hat{k}$,$\hat{j} + m \hat{k}$,and $m \hat{i} + \hat{k}$ becomes minimum when $m$ is

If the vectors $\bar{a}=\hat{\imath}-2 \hat{\jmath}+\hat{k}$,$\bar{b}=2 \hat{\imath}-5 \hat{\jmath}+p \hat{k}$ and $\bar{c}=5 \hat{\imath}-9 \hat{\jmath}+4 \hat{k}$ are coplanar,then the value of $p$ is

The four points whose position vectors are given by $2\bar{a}+3\bar{b}-\bar{c}$, $\bar{a}-2\bar{b}+3\bar{c}$, $3\bar{a}+4\bar{b}-2\bar{c}$ and $\bar{a}-6\bar{b}+6\bar{c}$ are

If the volume of a parallelepiped,whose coterminous edges are given by the vectors $\overrightarrow{a} = \hat{i} + \hat{j} + n\hat{k}$,$\overrightarrow{b} = 2\hat{i} + 4\hat{j} - n\hat{k}$,and $\overrightarrow{c} = \hat{i} + n\hat{j} + 3\hat{k}$ $(n \geq 0)$,is $158$ cubic units,then which of the following is true?

Let $\vec V = 2\hat i + \hat j - \hat k$,$\vec W = \hat i + 3\hat k$,and $|\vec U| = 2$. If $\vec U$ is a vector in the $x-y$ plane,then the greatest value of $([\vec U \vec V \vec W])^2$ is: