Let $\overrightarrow{a}=\hat{i}-2 \hat{j}+3 \hat{k}$, $\overrightarrow{b}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\overrightarrow{c}=\lambda \hat{i}+\hat{j}+(2 \lambda-1) \hat{k}$. If $\overrightarrow{c}$ is parallel to the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$, then $\lambda$ is equal to

If $u, v$ and $w$ are three non-coplanar vectors,then $(u + v - w) \cdot [(u - v) \times (v - w)]$ equals

$(\bar{a}+2 \bar{b}-\bar{c}) \cdot \{(\bar{a}-\bar{b}) \times (\bar{a}-\bar{b}-\bar{c})\} = $

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 the four points $A(6,2,4)$,$B(1,3,5)$,$C(1,-2,3)$,and $D(6, k, 2)$ are coplanar,then $k=$

If $\left[ {\vec a \,\vec b \,\vec c } \right] = 4$,then $\left[ {\vec a \times \vec b, \vec b \times \vec c, \vec c \times \vec a } \right] = \dots$