If $\bar{a} = 2\hat{i} + \hat{j} - \hat{k}$, $\bar{b} = \hat{i} + 3\hat{k}$ and $\bar{c}$ is a unit vector, then the maximum value of the scalar triple product $[\bar{a}\ \bar{b}\ \bar{c}]$ is

Let $\bar{a}, \bar{b}$ and $\bar{c}$ be three coplanar unit vectors. A unit vector $\bar{d}$ is perpendicular to them. If $(\bar{a} \times \bar{b}) \times (\bar{c} \times \bar{d}) = \dfrac{3}{26}\hat{i} - \dfrac{2}{13}\hat{j} + \dfrac{6}{13}\hat{k}$ and the angle between $\bar{a}$ and $\bar{b}$ is $30^\circ$, then $\bar{c}$ is equal to...

The volume of a parallelopiped with coterminous edges $\bar{a}, \bar{b}, \bar{c}$ is 3 cubic units. The volume (in cubic units) of a tetrahedron with coterminous edges $(\bar{a} \times \bar{b}), (\bar{a} \times 2\bar{c}), (\bar{b} \times 2\bar{c})$ is...

If the lines $2x = ky = -z$ and $6x = -y = -4z$ are perpendicular to each other then the value of $k$ is ...