If three vectors $2\hat{i}-\hat{j}-\hat{k}$,$\hat{i}+2\hat{j}-3\hat{k}$ and $3\hat{i}+\lambda\hat{j}+5\hat{k}$ are coplanar,then the value of $\lambda$ is

If $[\bar{a} \bar{b} \bar{c}] = 4$,then the volume (in cubic units) of the parallelepiped with $\bar{a} + 2 \bar{b}, \bar{b} + 2 \bar{c}$ and $\bar{c} + 2 \bar{a}$ as coterminal edges,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 $\bar{a} = \hat{i} - \hat{j}$,$\bar{b} = \hat{j} - \hat{k}$,and $\bar{c} = \hat{k} - \hat{i}$,then a unit vector $\bar{d}$ such that $\bar{a} \cdot \bar{d} = 0$ and $[\bar{b} \bar{c} \bar{d}] = 0$ is:

The volume of the parallelopiped whose coterminous edges are $\hat{j}+\hat{k}$, $\hat{i}+\hat{k}$, and $\hat{i}+\hat{j}$ is

Let $a=2 \hat{i}+\hat{j}-3 \hat{k}$ and $b=\hat{i}+3 \hat{j}+2 \hat{k}$. Then the volume of the parallelopiped having coterminous edges as $a, b$ and $c$, where $c$ is the vector perpendicular to the plane of $a, b$ and $|c|=2$ is