Let $V = 2\hat{i} + \hat{j} - \hat{k}$ and $W = \hat{i} + 3\hat{k}$. If $U$ is a unit vector, then the maximum value of $[U V W]$ is

Let $V = 2i + j - k$ and $W = i + 3k$. If $U$ is a unit vector,then the maximum value of the scalar triple product $[U V W]$ is

$i \cdot (j \times k) + j \cdot (k \times i) + k \cdot (i \times j) = $

The volume of the parallelepiped determined by the vectors $\vec{a} + \vec{b}, \vec{b} + \vec{c}$ and $\vec{c} + \vec{a}$ is $4$. Then the volume of the parallelepiped determined by the vectors $\vec{a} \times \vec{b}, \vec{b} \times \vec{c}$ and $\vec{c} \times \vec{a}$ is:

Let $\vec{p}=2 \hat{i}+\hat{j}+3 \hat{k}$ and $\vec{q}=\hat{i}-\hat{j}+\hat{k}$. If for some real numbers $\alpha, \beta$ and $\gamma$,we have $15 \hat{i}+10 \hat{j}+6 \hat{k}=\alpha(2 \vec{p}+\vec{q})+\beta(\vec{p}-2 \vec{q})+\gamma(\vec{p} \times \vec{q})$,then the value of $\gamma$ is.

What will be the volume of the parallelepiped whose coterminous edges are given by the vectors $a = i - j + k$,$b = i - 3j + 4k$,and $c = 2i - 5j + 3k$?