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 a vector $3\hat{i} + 4\hat{j} - 5\hat{k}$ is rotated through a certain angle about the origin in the anti-clockwise direction, then the components of the new vector are $a + 1, -3, 5$ . The possible values of $a$ is

Let A, B, C, D be the points in the plane with position vectors $-2\hat{i} - \hat{j}$, $4\hat{i}$, $3\hat{i} + 3\hat{j}$ and $-3\hat{i} + 2\hat{j}$ respectively, then $\square$ABCD is

Let $\bar{a} = \hat{i} + \hat{j}$, $\bar{c} = \hat{i} - \hat{j}$ and a vector $\bar{b}$ be such that $\bar{a} \times \bar{b} = \bar{c}$ and $\bar{a} \cdot \bar{b} = 3$ then $|\bar{b}| =$