The sum of all real values of $\lambda$ for which the vectors $\vec{a} = \lambda \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + \lambda \hat{j} + 2\hat{k}$, and $\vec{c} = 2\hat{i} + 3\hat{j} + \lambda \hat{k}$ are coplanar is:

If $\bar{a}$ is perpendicular to $\bar{b}$ and $\bar{c}$,$|\vec{a}|=2$,$|\bar{b}|=3$,$|\bar{c}|=4$ and the angle between $\bar{b}$ and $\bar{c}$ is $\frac{\pi}{3}$,then $\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]=$ (in $\sqrt{3}$)

If $\vec{w} = \alpha (\vec{a} \times \vec{b}) + \beta (\vec{b} \times \vec{c}) + \gamma (\vec{c} \times \vec{a})$,$[\vec{a}, \vec{b}, \vec{c}] = 2$ and $\vec{w} \cdot (\vec{a} + \vec{b} + \vec{c}) = 8$,then $\alpha + \beta + \gamma =$

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

Let $a = i - k$, $b = xi + j + (1 - x)k$, and $c = yi + xj + (1 + x - y)k$. Then $[a\,b\,c]$ depends on

Let $\overline{a}, \overline{b}, \overline{c}$ be three non-coplanar vectors and $\overline{p}, \overline{q}, \overline{r}$ be defined by the relations $\overline{p}=\frac{\overline{b} \times \overline{c}}{[\overline{a} \overline{b} \overline{c}]}, \overline{q}=\frac{\overline{c} \times \overline{a}}{[\overline{a} \overline{b} \overline{c}]}, \overline{r}=\frac{\overline{a} \times \overline{b}}{[\overline{a} \overline{b} \overline{c}]}$. Then the value of the expression $(\overline{a}+\overline{b}) \cdot \overline{p}+(\overline{b}+\overline{c}) \cdot \overline{q}+(\overline{c}+\overline{a}) \cdot \overline{r}$ is equal to: