If $[\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}] = \lambda [\vec{a}, \vec{b}, \vec{c}]^2$,then $\lambda$ is equal to:

If $\overrightarrow{a} = 2\hat{i} + \hat{j} + 3\hat{k}$,$\overrightarrow{b} = 3\hat{i} + 3\hat{j} + \hat{k}$ and $\overrightarrow{c} = c_{1}\hat{i} + c_{2}\hat{j} + c_{3}\hat{k}$ are coplanar vectors and $\overrightarrow{a} \cdot \overrightarrow{c} = 5$,$\overrightarrow{b} \perp \overrightarrow{c}$,then $122(c_{1} + c_{2} + c_{3})$ is equal to.......

If $2 \hat{i}-\hat{j}+3 \hat{k}$,$-12 \hat{i}-\hat{j}-3 \hat{k}$,$-\hat{i}+2 \hat{j}-4 \hat{k}$ and $\lambda \hat{i}+2 \hat{j}-\hat{k}$ are the position vectors of four coplanar points,then $\lambda=$

If $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$,$\vec{b} = 2\hat{i} + 3\hat{j} - \hat{k}$,and $\vec{c} = \lambda\hat{i} + \hat{j} + (2\lambda - 1)\hat{k}$ are coplanar vectors,then $\lambda$ is equal to:

If $a, b, c$ are three non-coplanar vectors,then $\frac{a \cdot (b \times c)}{c \times a \cdot b} + \frac{b \cdot (a \times c)}{c \cdot (a \times b)} = $

The volume of the tetrahedron formed by the vectors $\vec{a}, \vec{b}, \vec{c}$ 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: