For what value of $\lambda$ is the volume of the tetrahedron with vertices having position vectors $\hat{i} - 6\hat{j} + 10\hat{k}$,$-\hat{i} - 3\hat{j} + 7\hat{k}$,$5\hat{i} - \hat{j} + \lambda\hat{k}$,and $7\hat{i} - 4\hat{j} + 7\hat{k}$ equal to $11$ cubic units?

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 $[\vec{p} \vec{q} \vec{r}] = ...$

Which of the following expressions is meaningful?

Let $\vec{a} = (a_1\hat{i} + a_2\hat{j} + a_3\hat{k})$, $\vec{b} = (b_1\hat{i} + b_2\hat{j} + b_3\hat{k})$, and $\vec{c} = (c_1\hat{i} + c_2\hat{j} + c_3\hat{k})$ be three non-zero vectors such that $\vec{a}$ is a unit vector perpendicular to both $\vec{b}$ and $\vec{c}$. If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$, then find the value of $\left| \begin{matrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{matrix} \right|^2$.

Let $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - 2\hat{j} + 3\hat{k}$. For what value of $\lambda$ is the vector $\vec{c} = \lambda\hat{i} + \hat{j} + (2\lambda - 1)\hat{k}$ parallel to the plane containing $\vec{a}$ and $\vec{b}$?

If $\vec{w} = \alpha (\vec{a} \times \vec{b}) + \beta (\vec{b} \times \vec{c}) + \gamma (\vec{c} \times \vec{a})$,$[\vec{a}, \vec{b}, \vec{c}] = 2$ and $\vec{w} \cdot (\vec{a} + \vec{b} + \vec{c}) = 8$,then $\alpha + \beta + \gamma =$