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

Direction ratios of the line joining the points \((3,1,4)\) and \((7,2,12)\) are
\(
\begin{array}{l}
=\langle 7-3,2-1,12-4\rangle \\
= < 4,1,8> \\
=\left\langle a_{1}, a_{2}, a_{3}\right\rangle
\end{array}
\)
And the direction ratio of given line is
\(
\begin{array}{l}
= < 2,2,1> \\
= < b_{1}, b_{2}, b_{3}>
\end{array}
\)
Let \(Q\) be the angle between the lines,
\(
\text { then } \cos \theta=\frac{a_{1} b_{1}+a_{2} b_{2}+a_{3} b_{3}}{\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}} \sqrt{b_{1}^{2}+b_{2}^{2}+b_{3}^{2}}}
\)
\(\Rightarrow \cos \theta=\frac{(4)(2)+(1)(2)+(8)(1)}{\sqrt{16+1+64} \sqrt{4+4+1}} \)
\( \Rightarrow \cos \theta=\frac{18}{\sqrt{81} \sqrt{9}}=\frac{18}{9 \times 3}=\frac{2}{3} \)
\( \Rightarrow \theta=\cos ^{-1}\left(\frac{2}{3}\right)\)