The volume of a tetrahedron with vertices $5\hat{i} - \hat{j} + \hat{k}, 7\hat{i} - 4\hat{j} + p\hat{k}, \hat{i} - 6\hat{j} + 10\hat{k}$ and $-\hat{i} - 3\hat{j} + 7\hat{k}$ is 11 cubic units, then one of the values of p is

Let $\bar{a}, \bar{b}, \bar{c}$ be unit vectors such that $\bar{a}$ is perpendicular to the plane of $\bar{b}$ and $\bar{c}$. If the angle between $\bar{b}$ and $\bar{c}$ is $\dfrac{\pi}{3}$, then $|\bar{a} + \bar{b} + \bar{c}| =$

If a parallelogram is constructed on the vectors $\bar{a} = 3\bar{p} - \bar{q}, \bar{b} = \bar{p} + 3\bar{q}$ and $|\bar{p}| = 3, |\bar{q}| = 2$ and angle between $\bar{p}$ and $\bar{q}$ is $\dfrac{\pi}{3}$, then the ratio of the lengths of adjacent sides $\bar{a}$ and $\bar{b}$ of the parallelogram is

A vector $\bar{r}$ of magnitude $3\sqrt{2}$ units which makes angles of $\dfrac{\pi}{4}$ and $\dfrac{\pi}{2}$ respectively with Y and Z axes is