If the volume of the parallelepiped with $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ as coterminous edges is $40 \text{ cubic units}$,then the volume of the parallelepiped having $\overrightarrow{b}+\overrightarrow{c}, \overrightarrow{c}+\overrightarrow{a}$ and $\overrightarrow{a}+\overrightarrow{b}$ as coterminous edges in cubic units is

Let $O$ be the origin. Let $\overline{OP} = x\hat{i} + y\hat{j} - \hat{k}$ and $\overline{OQ} = -\hat{i} + 2\hat{j} + 3x\hat{k}$,where $x, y \in \mathbb{R}$ and $x > 0$,be such that $|\overline{PQ}| = \sqrt{20}$ and the vector $\overline{OP}$ is perpendicular to $\overline{OQ}$. If $\overline{OR} = 3\hat{i} + z\hat{j} - 7\hat{k}$,where $z \in \mathbb{R}$,is coplanar with $\overline{OP}$ and $\overline{OQ}$,then the value of $x^2 + y^2 + z^2$ is equal to ...... .

If $a, b$ and $c$ are non-coplanar vectors and the four points with position vectors $2a+3b-c$,$a-2b+3c$,$3a+4b-2c$ and $ka-6b+6c$ are coplanar,then $k=$

If the vectors $ai + j + k$,$i + bj + k$,and $i + j + ck$ $(a \ne 1, b \ne 1, c \ne 1)$ are coplanar,then the value of $\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} = $

If $a, b$ and $c$ are three non-coplanar vectors and $p, q$ and $r$ are vectors defined by $p=\frac{b \times c}{[a b c]}, q=\frac{c \times a}{[a b c]}, r=\frac{a \times b}{[a b c]}$,then $(a+b) \cdot p+(b+c) \cdot q+(c+a) \cdot r$ 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: