Let $p, q, r$ be three non-coplanar vectors and $b = p \times q$. If $a, b, c$ denote the coterminous edges of a parallelepiped, then its height with the base having $a$ and $c$ is

If $2i + 3j$,$i + j + k$,and $\lambda i + 4j + 2k$ taken in an order are coterminous edges of a parallelepiped of volume $2$ cubic units,then the value of $\lambda$ is:

If $\bar{a} = \hat{i} - \hat{j}$,$\bar{b} = \hat{j} - \hat{k}$,and $\bar{c} = \hat{k} - \hat{i}$,then a unit vector $\bar{d}$ such that $\bar{a} \cdot \bar{d} = 0$ and $[\bar{b} \bar{c} \bar{d}] = 0$ is:

$a \cdot (b \times c)$ is equal to

The value of $m$,if the vectors $\hat{\imath}-\hat{\jmath}-6 \hat{k}$,$\hat{\imath}-3 \hat{\jmath}+4 \hat{k}$,and $2 \hat{\imath}-5 \hat{\jmath}+m \hat{k}$ are coplanar,is

Let $\overline{a}, \overline{b}, \overline{c}$ be three non-coplanar vectors and $\overline{p}, \overline{q}, \overline{r}$ be defined by the relations $\overline{p}=\frac{\overline{b} \times \overline{c}}{[\overline{a} \overline{b} \overline{c}]}, \overline{q}=\frac{\overline{c} \times \overline{a}}{[\overline{a} \overline{b} \overline{c}]}, \overline{r}=\frac{\overline{a} \times \overline{b}}{[\overline{a} \overline{b} \overline{c}]}$. Then the value of the expression $(\overline{a}+\overline{b}) \cdot \overline{p}+(\overline{b}+\overline{c}) \cdot \overline{q}+(\overline{c}+\overline{a}) \cdot \overline{r}$ is equal to: