Let $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$, $\vec{b} = 3\hat{i} + 2\hat{j} + 2\hat{k}$, and $\vec{c} = 4\hat{i} - 3\hat{j} + \hat{k}$. The vectors $\vec{a}, \vec{b}, \text{ and } \vec{c}$ are:

Volume of a parallelepiped with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is $12$ cubic units. The volume of a tetrahedron with coterminous edges $\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{a} + \vec{b} - \vec{c}$ will be ............. cubic units.

Let $\bar{a}$ and $\bar{c}$ be unit vectors at an angle $\frac{\pi}{3}$ with each other. If $(\bar{a} \times(\bar{b} \times \bar{c})) \cdot(\bar{a} \times \bar{c})=5$,then $\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]=$

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 $\vec{c}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$. If the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{6}$,then $\left| \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{array} \right|^2 = \dots$

Let $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$,$\vec{b} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{c}$ be a vector such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$. If $\vec{c} = x\vec{a} + y\vec{b} + z(\vec{a} \times \vec{b})$,then the value of $x + y + z$ is:

Value of $\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{i} \cdot(\hat{j} \times \hat{j})+\hat{k} \cdot(\hat{j} \times \hat{i})+\hat{i} \cdot(\hat{k} \times \hat{j})$ is . . . . . . .