The maximum volume of a parallelepiped (in cubic units) with vectors $\vec{u} = 2a\hat{i} + \hat{k}$, $\vec{v} = a\hat{j} - a\hat{k}$, and $\vec{w} = 3\hat{i} + a\hat{j}$, where $a \in [0, 1]$, as its coterminous edges is:

For any non-zero vectors $\bar{a}$ and $\bar{b}$,$\left[\begin{array}{lll}\bar{b} & \bar{a} \times \bar{b} & \bar{a}\end{array}\right]=$

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three non-coplanar vectors and $\vec{p}, \vec{q}$, and $\vec{r}$ are defined by $\vec{p}=\frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}, \vec{q}=\frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]}, \vec{r}=\frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]}$, then find the value of $(\vec{a}+\vec{b}) \cdot \vec{p} + (\vec{b}+\vec{c}) \cdot \vec{q} + (\vec{c}+\vec{a}) \cdot \vec{r}$.

If $|\vec{a}| = 4, |\vec{b}| = 3$ and $\vec{a} \cdot \vec{b} = 8$, then the scalar triple product $[\vec{a} \quad \vec{a} + \vec{b} \quad \vec{a} \times \vec{b}]$ is equal to:

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

The volume of the tetrahedron,whose vertices are given by the vectors $-i + j + k$,$i - j + k$,and $i + j - k$ with reference to the fourth vertex as the origin,is