The volume of the parallelepiped whose coterminous edges are $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$, $\vec{b} = 3\hat{i} - \hat{j} - \hat{k}$, and $\vec{c} = \hat{j} + 3\hat{k}$ is:

The volume of the tetrahedron formed by the coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is $3$. Then the volume of the parallelepiped formed by the coterminous edges $\vec{a} + \vec{b}, \vec{b} + \vec{c}, \vec{c} + \vec{a}$ is

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 $\left| \begin{matrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{matrix} \right|^2$.

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{b} = 2\hat{i} - 4\hat{k}$,and $\vec{c} = \hat{i} + \lambda \hat{j} + 3\hat{k}$ are coplanar,then the value of $\lambda$ is:

If $2i + 3j$,$i + j + k$,and $\lambda i + 4j + 2k$ taken in an order are coterminous edges of a parallelepiped of volume $2$ cubic units,then the value of $\lambda$ is:

Let $S$ be the set of all $(\lambda, \mu)$ for which the vectors $\lambda \hat{i} - \hat{j} + \hat{k}$,$\hat{i} + 2\hat{j} + \mu \hat{k}$ and $3\hat{i} - 4\hat{j} + 5\hat{k}$,where $\lambda - \mu = 5$,are coplanar,then $\sum_{(\lambda, \mu) \in S} 80(\lambda^2 + \mu^2)$ is equal to :