Let $\bar{a}=\lambda \bar{i}+3 \bar{j}+4 \bar{k}$,$\bar{b}=3 \bar{i}-\bar{j}+\lambda \bar{k}$ and $\bar{c}=\lambda \bar{i}+\bar{j}-3 \bar{k}$ be three vectors for some integer $\lambda$. If the volume of the parallelepiped with $\bar{a}, \bar{b}, \bar{c}$ as coterminus edges is $61$ cubic units,then the number of possible values of $\lambda$ is

If $\overrightarrow{a} = \alpha \hat{i} + \beta \hat{j} + 3 \hat{k}$,$\overrightarrow{b} = -\beta \hat{i} - \alpha \hat{j} - \hat{k}$,and $\overrightarrow{c} = \hat{i} - 2 \hat{j} - \hat{k}$ such that $\overrightarrow{a} \cdot \overrightarrow{b} = 1$ and $\overrightarrow{b} \cdot \overrightarrow{c} = -3$,then $\frac{1}{3}((\vec{a} \times \vec{b}) \cdot \vec{c})$ 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

Let $\vec{a}=\hat{i}-\alpha \hat{j}+\beta \hat{k}$,$\vec{b}=3 \hat{i}+\beta \hat{j}-\alpha \hat{k}$ and $\vec{c}=-\alpha \hat{i}-2 \hat{j}+\hat{k}$,where $\alpha$ and $\beta$ are integers. If $\vec{a} \cdot \vec{b}=-1$ and $\vec{b} \cdot \vec{c}=10$,then $(\vec{a} \times \vec{b}) \cdot \vec{c}$ is equal to $.....$

If the volume of a tetrahedron having $\bar{i}+2 \bar{j}-3 \bar{k}$,$2 \bar{i}+\bar{j}-3 \bar{k}$,and $3 \bar{i}-\bar{j}+p \bar{k}$ as its coterminous edges is $2$,then the values of $p$ are the roots of the equation

$[a, b, a \times b]$ is equal to