$[(\vec{a} \times \vec{b}) \times (\vec{a} \times \vec{c})] \cdot \vec{d} = \dots$

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

If the points whose position vectors are $3i - 2j - k,$ $2i + 3j - 4k,$ $-i + j + 2k,$ and $4i + 5j + \lambda k$ lie on a plane,then $\lambda = $

$[(a \times b) \times (b \times c), (b \times c) \times (c \times a), (c \times a) \times (a \times b)] = \,$

If the vectors $a\hat{i}+\hat{j}+\hat{k}$,$\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ are coplanar $(a \neq 1, b \neq 1, c \neq 1)$,then the value of $abc-(a+b+c)$ is:

Let $\overrightarrow{PR}=3 \hat{i}+\hat{j}-2 \hat{k}$ and $\overrightarrow{SQ}=\hat{i}-3 \hat{j}-4 \hat{k}$ be the diagonals of a parallelogram $PQRS$,and let $\overrightarrow{PT}=\hat{i}+2 \hat{j}+3 \hat{k}$ be another vector. Then the volume of the parallelepiped determined by the vectors $\overrightarrow{PT}, \overrightarrow{PQ}$ and $\overrightarrow{PS}$ is: