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

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 $\vec{u} = \hat{i} + 2\hat{j} - 2\hat{k}$, $\vec{v} = 2\hat{i} + \hat{k}$ and $\vec{w}$ is a unit vector, then the maximum value of the scalar triple product $[\vec{u} \vec{v} \vec{w}]$ is

If $a, b$ and $c$ are non-coplanar,then the value of $a \cdot \left\{ \frac{b \times c}{3 b \cdot (c \times a)} \right\} - b \cdot \left\{ \frac{c \times a}{2 c \cdot (a \times b)} \right\}$ is

Consider $\overrightarrow{r}, \overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ are non-zero vectors such that $\overrightarrow{r} \cdot \overrightarrow{a}=0$,$|\overrightarrow{r} \times \overrightarrow{b}|=|\overrightarrow{r}||\overrightarrow{b}|$,and $|\overrightarrow{r} \times \overrightarrow{c}|=|\overrightarrow{r}||\overrightarrow{c}|$. Then,the scalar triple product $[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$ is:

If the four points with position vectors $-2\hat{i}+\hat{j}+\hat{k}$, $\hat{i}+\hat{j}+\hat{k}$, $\hat{j}-\hat{k}$, and $\lambda\hat{j}+\hat{k}$ are coplanar, then $\lambda$ is equal to