If $\vec{a}=2 \hat{i}-\hat{j}+3 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}+\hat{k}$,and $\vec{c}=3 \hat{i}-\hat{j}+2 \hat{k}$,then $\vec{a} \cdot(\vec{b} \times \vec{c})=$ . . . . . . .

The altitude of the parallelepiped, whose coterminous edges are the vectors $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + 4\hat{j} - \hat{k}$, and $\vec{c} = \hat{i} + \hat{j} + 3\hat{k}$, where $\vec{a}$ and $\vec{b}$ are the sides of the base of the parallelepiped, is

If the vectors $p \hat{i}+\hat{j}+\hat{k}$,$\hat{i}+q \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+r \hat{k}$ $(p \neq q \neq r \neq 1)$ are coplanar,then the value of $pqr-(p+q+r)$ is

If $\bar{a}=3 \hat{i}+\hat{j}-\hat{k}, \bar{b}=2 \hat{i}-\hat{j}+23 \hat{k}$ and $\bar{c}=7 \hat{i}-\hat{j}+23 \hat{k}$,then which of the following is valid?

If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors and $\vec{p}, \vec{q}, \vec{r}$ are defined as $\vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}, \vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]}, \vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]}$, then $[(\vec{a} + \vec{b}) \cdot \vec{p} + (\vec{b} + \vec{c}) \cdot \vec{q} + (\vec{c} + \vec{a}) \cdot \vec{r}]$ is equal to

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