If $a=2u+3v+7w$,$b=u+v-2w$ and $c=-u-2v-3w$,then $\left|\frac{[u, v, w]}{[a, b, c]}\right|(a+b+c) = $

If $\vec{a}, \vec{b}, \vec{c}$ are any three non-zero non-coplanar vectors and vectors $\vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}, \vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]}, \vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]}$,then $[\vec{p} \vec{q} \vec{r}] = ...$

If the points whose position vectors are $3i - 2j - k,$ $2i + 3j - 4k,$ $-i + j + 2k,$ and $4i + 5j + \lambda k$ lie on a plane,then $\lambda = $

The altitude of the parallelepiped,whose coterminus edges are the vectors $\bar{a}=\hat{i}+\hat{j}+\hat{k}$,$\bar{b}=2\hat{i}+4\hat{j}-\hat{k}$,and $\bar{c}=\hat{i}+\hat{j}+3\hat{k}$,where $\bar{a}$ and $\bar{b}$ are the sides of the base of the parallelepiped,is:

If $\vec{a} = 4\hat{i} - 2\hat{j} + \hat{k}$,$\vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}$,and $\vec{c} = 2\hat{i} - \hat{j} + 2\hat{k}$ represent the three coterminous edges of a parallelepiped,find its volume.

Let $p, q, r$ be three non-coplanar vectors and $b = p \times q$. If $a, b, c$ denote the coterminous edges of a parallelepiped,then its height with the base having $a$ and $c$ is