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

A plane meets the co-ordinate axes in A, B, C such that the centroid of the triangle ABC is the point $(1, r, r^2)$, then the equation of the plane is,

If M denotes the midpoint of the line joining A$(4, 5, -10)$ and B$(-1, 2, 1)$, then the equation of the plane through M and perpendicular to AB is: