If the vectors $m \hat{i} + m \hat{j} + n \hat{k}$,$\hat{i} + \hat{k}$,and $n \hat{i} + n \hat{j} + p \hat{k}$ lie in a plane,then...

Let $a=p(\hat{i}+\hat{j}+\hat{k})$,$b=\hat{i}+\hat{j}-2\hat{k}$,and $c=2\hat{i}-\hat{j}+2\hat{k}$ be three vectors. If the value of $[abc]$ is not more than $15$ and not less than $-5$,then $p$ lies in the interval:

If $a = -3i + 7j + 5k$,$b = -3i + 7j - 3k$,and $c = 7i - 5j - 3k$ are the three coterminous edges of a parallelepiped,then its volume is

If $a = i + j + k$,$b = 4i + 3j + 4k$,and $c = i + \alpha j + \beta k$ are coplanar vectors and $|c| = \sqrt{3}$,then:

If the volume of a tetrahedron having $\bar{i}+2 \bar{j}-3 \bar{k}$,$2 \bar{i}+\bar{j}-3 \bar{k}$,and $3 \bar{i}-\bar{j}+p \bar{k}$ as its coterminous edges is $2$,then the values of $p$ are the roots of the equation

If $\bar{a}, \bar{b}, \bar{c}$ are nonzero vectors along the coterminus edges of a parallelepiped with volume $7$ cubic units,then the volume of a parallelepiped with $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$ as the coterminus edges is