Let $\overrightarrow{a}$ be a unit vector,$\overrightarrow{b} = 2\hat{i} + \hat{j} - \hat{k}$ and $\overrightarrow{c} = \hat{i} + 3\hat{k}$. Then,the maximum value of $[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$ is

The volume of the tetrahedron whose co-terminus edges are $\bar{a}, \bar{b}, \bar{c}$ is $12$ cubic units. If the scalar projection of $\bar{a}$ on $\bar{b} \times \bar{c}$ is $4$,then $|\bar{b} \times \bar{c}|=$

If $\alpha = 2i + 3j - k$,$\beta = -i + 2j - 4k$,and $\gamma = i + j + k$,then $(\alpha \times \beta) \cdot (\alpha \times \gamma)$ is equal to

Let $O$ be the origin. Let $\overline{OP} = x\hat{i} + y\hat{j} - \hat{k}$ and $\overline{OQ} = -\hat{i} + 2\hat{j} + 3x\hat{k}$,where $x, y \in \mathbb{R}$ and $x > 0$,be such that $|\overline{PQ}| = \sqrt{20}$ and the vector $\overline{OP}$ is perpendicular to $\overline{OQ}$. If $\overline{OR} = 3\hat{i} + z\hat{j} - 7\hat{k}$,where $z \in \mathbb{R}$,is coplanar with $\overline{OP}$ and $\overline{OQ}$,then the value of $x^2 + y^2 + z^2$ is equal to ...... .

If $a, b, c$ are any three vectors and their reciprocal vectors are $a^{-1}, b^{-1}, c^{-1}$ such that $[a, b, c] \neq 0$,then $[a^{-1}, b^{-1}, c^{-1}]$ is equal to:

Let the volume of tetrahedron $ABCD$ be $81$ cubic units and $G_1, G_2, G_3$ be the centroids of the triangular faces $ABC, ABD,$ and $ACD$ respectively. Then the volume of tetrahedron $AG_1G_2G_3$ is (in cubic units):