Let $V = 2\hat{i} + \hat{j} - \hat{k}$ and $W = \hat{i} + 3\hat{k}$. If $U$ is a unit vector, then the maximum value of $[U V W]$ is

Evaluate: $\vec{a} \cdot \{(\vec{b} + \vec{c}) \times (\vec{a} + \vec{b} + \vec{c})\}$

The lines $\overline{r}=\overline{a}+\lambda(\overline{b} \times \overline{c})$ and $\overline{r}=\overline{c}+\mu(\overline{a} \times \overline{b})$ will intersect if

$i \cdot (j \times k) + j \cdot (k \times i) + k \cdot (i \times j) = $

The volume of the parallelepiped whose coterminous edges are $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$, $\vec{b} = 3\hat{i} - \hat{j} - \hat{k}$, and $\vec{c} = \hat{j} + 3\hat{k}$ is:

If $\overrightarrow{a} = 2\hat{i} + \hat{j} + 3\hat{k}$,$\overrightarrow{b} = 3\hat{i} + 3\hat{j} + \hat{k}$ and $\overrightarrow{c} = c_{1}\hat{i} + c_{2}\hat{j} + c_{3}\hat{k}$ are coplanar vectors and $\overrightarrow{a} \cdot \overrightarrow{c} = 5$,$\overrightarrow{b} \perp \overrightarrow{c}$,then $122(c_{1} + c_{2} + c_{3})$ is equal to.......