Let $\vec{a} = 2\hat{i} + \hat{k}, \vec{b} = \hat{i} + \hat{j} + \hat{k}$, and $\vec{c} = 4\hat{i} - 3\hat{j} + 7\hat{k}$. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{b} = \vec{c} \times \vec{b}$ and $\vec{r} \cdot \vec{a} = 0$, Then $\vec{r} \cdot \vec{c} =$

If $\vec{u} = \hat{i} + 2\hat{j} - 2\hat{k}, \vec{v} = 2\hat{i} + \hat{k}$ and $\vec{w}$ is unit vector then the maximum value of scalar triple product $[\vec{u}\ \vec{v}\ \vec{w}]$ is

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