If $\bar{a}$ and $\bar{b}$ are unit vectors perpendicular to each other, then $\left[\bar{a} + (\bar{a} \times \bar{b})\quad \bar{b} + (\bar{a} \times \bar{b})\quad (\bar{a} \times \bar{b})\right] = \cdots$

If $\bar{a}$, $\bar{b}$ and $\bar{c}$ are three vectors such that $|\bar{a} + \bar{b} + \bar{c}| = 1$, $\bar{c} = \lambda(\bar{a} \times \bar{b})$ and $|\bar{a}| = \dfrac{1}{\sqrt{3}}$, $|\bar{b}| = \dfrac{1}{\sqrt{2}}$, $|\bar{c}| = \dfrac{1}{\sqrt{6}}$, then the angle between $\bar{a}$ and $\bar{b}$ is

The acute angle between the line $\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})$ and the plane $\vec{r} \cdot (2\hat{i} + p\hat{j} + \hat{k}) = 8$ is $\sin^{-1}\left(\dfrac{\sqrt{2}}{3}\right)$, then the value of $p$ are...

The shortest distance between the lines $\dfrac{x - 3}{3} = \dfrac{y - 8}{-1} = \dfrac{z - 3}{1}$ and $\dfrac{x + 3}{-3} = \dfrac{y + 7}{2} = \dfrac{z - 6}{4}$ is