If the volume of the parallelepiped formed by three non-coplanar vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $4$ cubic units,then $[\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]$ is equal to

If the vectors $i + 3j$,$5k$,and $\Pi - j$ are coplanar,find the value of $P$.

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 $\overline{p}=\hat{i}+\hat{j}+\hat{k}$ and $\overline{q}=\hat{i}-2 \hat{j}+\hat{k}$. Then a vector of magnitude $5 \sqrt{3}$ units perpendicular to the vector $\overline{q}$ and coplanar with $\overline{p}$ and $\overline{q}$ is

If the volume of a parallelepiped with coterminous edges $4 \hat{i} + 5 \hat{j} + \hat{k}$, $-\hat{j} + \hat{k}$, and $3 \hat{i} + 9 \hat{j} + p \hat{k}$ is $34$ cubic units, then $p$ is equal to:

Let $\overrightarrow{a}=\hat{i}-2 \hat{j}+3 \hat{k}$, $\overrightarrow{b}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\overrightarrow{c}=\lambda \hat{i}+\hat{j}+(2 \lambda-1) \hat{k}$. If $\overrightarrow{c}$ is parallel to the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$, then $\lambda$ is equal to