Let $a, b$ and $c$ be distinct positive numbers. If the vectors $a \hat{i} + a \hat{j} + c \hat{k}$,$\hat{i} + \hat{k}$ and $c \hat{i} + c \hat{j} + b \hat{k}$ are coplanar,then $c$ is equal to:

If $a = -3i + 7j + 5k$,$b = -3i + 7j - 3k$,and $c = 7i - 5j - 3k$ are the three coterminous edges of a parallelepiped,then its volume is

If $a, b, c$ are non-coplanar vectors and $\lambda$ is a real number,then the vectors $a + 2b + 3c, \lambda b + 4c$ and $(2\lambda - 1)c$ are non-coplanar for

If the volume of the parallelepiped formed by three non-coplanar vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $4$ cubic units,then $[\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]$ is equal to

If $a, b,$ and $c$ are coplanar unit vectors,find the value of the scalar triple product $[2a - b, 2b - c, 2c - a]$.

If the volume of the parallelepiped with $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ as coterminous edges is $40 \text{ cubic units}$,then the volume of the parallelepiped having $\overrightarrow{b}+\overrightarrow{c}, \overrightarrow{c}+\overrightarrow{a}$ and $\overrightarrow{a}+\overrightarrow{b}$ as coterminous edges in cubic units is