If $\vec{a} = 4\hat{i} - 2\hat{j} + \hat{k}$,$\vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}$,and $\vec{c} = 2\hat{i} - \hat{j} + 2\hat{k}$ represent the three coterminous edges of a parallelepiped,find its volume.

The volume of the tetrahedron formed by the vectors $\vec{a}, \vec{b}, \vec{c}$ is $3$. Then the volume of the parallelepiped formed by the coterminous edges $\vec{a} + \vec{b}, \vec{b} + \vec{c}, \vec{c} + \vec{a}$ is:

If $\overline{a}$ and $\overline{c}$ are unit vectors inclined at $\frac{\pi}{3}$ with each other and $(\overline{a} \times (\overline{b} \times \overline{c})) \cdot (\overline{a} \times \overline{c}) = 5$,then the value of $5[\overline{a} \overline{b} \overline{c}] = $

Let the vectors $\vec{u} = (2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k}$,$\vec{v} = (1+b) \hat{i}+2 b \hat{j}-b \hat{k}$,and $\vec{w} = (2+b) \hat{i}+2 b \hat{j}+(1-b) \hat{k}$ where $a, b, c \in \mathbb{R}$ be co-planar. Then which of the following is true?

If the vectors $\hat{\imath}+\hat{\jmath}+\hat{k}$,$\hat{\imath}-\hat{\jmath}+\hat{k}$ and $2\hat{\imath}+3\hat{\jmath}+m\hat{k}$ are coplanar,then $m=$

Let the volume of a parallelepiped whose coterminous edges are given by $\overrightarrow{u}=\hat{i}+\hat{j}+\lambda \hat{k}$,$\overrightarrow{v}=\hat{i}+\hat{j}+3 \hat{k}$ and $\overrightarrow{w}=2 \hat{i}+\hat{j}+\hat{k}$ be $1 \text{ cu. unit}$. If $\theta$ is the angle between the edges $\overrightarrow{u}$ and $\overrightarrow{w}$,then $\cos \theta$ can be