If the given vectors $(-bc, b^2 + bc, c^2 + bc)$,$(a^2 + ac, -ac, c^2 + ac)$ and $(a^2 + ab, b^2 + ab, -ab)$ are coplanar,where none of $a, b$ and $c$ is zero,then:

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=2\hat{i}+\lambda\hat{j}+\hat{k}$,$\vec{c}=\hat{i}-\hat{j}+4\hat{k}$ and $\vec{a} \cdot (\vec{b} \times \vec{c}) = 10$,then $\lambda$ is equal to

If $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$ are coterminous edges of a parallelepiped,then its volume 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 $\sqrt{3}$)

$\bar{a} = \hat{i} + \hat{j} + \hat{k}$,$\bar{b} = \hat{i} - \hat{j} + 2\hat{k}$ and $\bar{c} = x\hat{i} + (x-1)\hat{j} - \hat{k}$. If the vector $\bar{c}$ lies in the plane of $\bar{a}$ and $\bar{b}$,then $x=$

If $a=x \hat{i}+y \hat{j}+z \hat{k}$, then $(a \times \hat{i}) \cdot(\hat{i}+\hat{j})+(a \times \hat{j}) \cdot(\hat{j}+\hat{k})+(a \times \hat{k}) \cdot(\hat{k}+\hat{i})=$