The angle between a line with direction ratios proportional to 2, 2, 1 and a line joining (3, 1, 4) to (7, 2, 12), is

$cos^{-1}\left(\frac{2}{3}\right)$

A line with direction ratios proportional to 2, 2, 1 is parallel to the vector $\vec{a} = 2 \hat{i} + 2 \hat{j} + \hat{k}.$

Line joining P(3, 1, 4) to Q( 7, 2, 12) is parallel to the vector

$\vec{PQ}=4\hat{i} + \hat{j} + 8 \hat{k} $

Let $\theta $ be the required angle. Then,

$cos theta = \frac{\vec{a}.\vec{PQ}}{|\vec{a}||\vec{PQ}|}=\frac{8+2+8}{\sqrt{4+4+1}\sqrt{16+1+64}}$

$ cos theta = \frac{18}{3× 9} = \frac{2}{3}⇒ \theta = cos^{-1}\left(\frac{2}{3}\right)$