The angle between the line $\bar{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})$ and the plane $\bar{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 8$ is $\ldots$

The acute angle between the lines $2x = 3y = -z$ and $6x = -y = -4z$ is $\ldots$

The line $\ell$ passes through the point $(2, 1, 1)$ and is parallel to the plane $x + y + 2z = 18$. If line $\ell$ intersects the line $\dfrac{x+2}{3} = \dfrac{y+1}{-1} = \dfrac{z-2}{1}$, then equation of the line $\ell$ is...

The acute angle $\theta$ between the $xy$-plane and the plane passing through the point $(1, 2, 4)$ and parallel to the vectors with direction ratios $3, 2, -1$ and $1, -2, -2$ is...