If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors and $\vec{p}, \vec{q}, \vec{r}$ are defined as $\vec{p} = \dfrac{\vec{b} \times \vec{c}}{[\vec{a}\ \vec{b}\ \vec{c}]}, \vec{q} = \dfrac{\vec{c} \times \vec{a}}{[\vec{a}\ \vec{b}\ \vec{c}]}, \vec{r} = \dfrac{\vec{a} \times \vec{b}}{[\vec{a}\ \vec{b}\ \vec{c}]}$, 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 volume of the parallelopiped whose coterminous edges are $2\hat{i} + \hat{j} - \hat{k}, 3\hat{i} - \hat{j} - \hat{k}, \hat{j} + 3\hat{k}$ is

Given the following expression
A) $(\vec{a} \times \vec{b}) \cdot \vec{c}$
B) $\vec{a} \times (\vec{b} \cdot \vec{c})$
C) $\vec{a} \cdot (\vec{b} \cdot \vec{c})$
D) $|\vec{a}|(\vec{b} \cdot \vec{c})$
E) $(\vec{a} \cdot \vec{b}) \times (\vec{b} \cdot \vec{c})$
Then which of the following is not correct

Let $\bar{a} = \hat{i} + \hat{j} + \hat{k}$, $\bar{b} = \hat{i} - 3\hat{j} + 2\hat{k}$ and $\bar{c} = 3\hat{i} - 2\hat{k}$. If a vector $\bar{p}$ satisfies the conditions $\bar{p} \cdot \bar{c} = 0$ and $\bar{p} \times \bar{a} = \bar{b} \times \bar{a}$, then the value of $|\bar{p}| = $...