Let $a=p(\hat{i}+\hat{j}+\hat{k})$,$b=\hat{i}+\hat{j}-2\hat{k}$,and $c=2\hat{i}-\hat{j}+2\hat{k}$ be three vectors. If the value of $[abc]$ is not more than $15$ and not less than $-5$,then $p$ lies in the interval:

If $\bar{a}, \bar{b}$ and $\bar{c}$ are any three non-zero vectors,then $(\bar{a}+2 \bar{b}+\bar{c}) \cdot[(\bar{a}-\bar{b}) \times(\bar{a}-\bar{b}-\bar{c})]=$

If the vectors $2i - j + k$,$i + 2j - 3k$,and $3i + \lambda j + 5k$ are coplanar,then $\lambda = $

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A, B, C$ and $D$ be $\vec{a}-\vec{b}+\vec{c}$,$\lambda \vec{a}-3 \vec{b}+4 \vec{c}$,$-\vec{a}+2 \vec{b}-3 \vec{c}$ and $2 \vec{a}-4 \vec{b}+6 \vec{c}$ respectively. If $\overrightarrow{AB}$,$\overrightarrow{AC}$ and $\overrightarrow{AD}$ are coplanar,then $\lambda$ is :

For what value of $\lambda$ are the vectors $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$,$\vec{b} = \lambda\hat{i} + 4\hat{j} + 7\hat{k}$,and $\vec{c} = -3\hat{i} - 2\hat{j} - 5\hat{k}$ coplanar?

$i \cdot (j \times k) + j \cdot (k \times i) + k \cdot (i \times j) = $