Consider the set of eight vectors $V=\{a \hat{i}+b \hat{j}+c \hat{k}: a, b, c \in\{-1,1\}\}$. Three non-coplanar vectors can be chosen from $V$ in $2^p$ ways. Then $p$ is

If the volume of a parallelepiped,whose coterminous edges are given by the vectors $\overrightarrow{a} = \hat{i} + \hat{j} + n\hat{k}$,$\overrightarrow{b} = 2\hat{i} + 4\hat{j} - n\hat{k}$,and $\overrightarrow{c} = \hat{i} + n\hat{j} + 3\hat{k}$ $(n \geq 0)$,is $158$ cubic units,then which of the following is true?

If the vertices of a tetrahedron are $\vec{a} = \vec{j} + 2\vec{k}$,$\vec{b} = 3\vec{i} + \vec{k}$,$\vec{c} = 4\vec{i} + 3\vec{j} + 6\vec{k}$,and $\vec{d} = 2\vec{i} + 3\vec{j} + 2\vec{k}$,find its volume.

Let $\vec{a}, \vec{b}, \vec{c}$ be unit vectors such that $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = \cos \theta$. Then the maximum value of $\theta$ is,where $\theta \in [0, \pi]$.

If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors and $\vec{r}$ is an arbitrary vector,then $(\vec{a} \times \vec{b}) \times (\vec{r} \times \vec{c}) + (\vec{b} \times \vec{c}) \times (\vec{r} \times \vec{a}) + (\vec{c} \times \vec{a}) \times (\vec{r} \times \vec{b}) = \dots$

Let $a = i - j$,$b = j - k$,$c = k - i$. If $\hat{d}$ is a unit vector such that $a \cdot \hat{d} = 0$ and $[b, c, \hat{d}] = 0$,then $\hat{d}$ is equal to