The angle between the lines whose direction cosines are \(\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\right)\) and \(\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2}\right)\) is

Given, direction cosines of line \(1:\left(l_{1}, m_{1}, n_{1}\right)\)
\(=\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\right)\)
direction cosines of line \(2 \Rightarrow\left(l_{2}, m_{2}, n_{2}\right)\)
\(=\left(\frac{\sqrt{3}}{4}, \frac{1}{4},-\frac{\sqrt{3}}{2}\right)\)
We know that, angle between two lines is given by \(\cos \theta=\left|l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}\right|\)
\(\begin{aligned}
\cos \theta &=\left|\frac{\sqrt{3}}{4} \times \frac{\sqrt{3}}{4}+\frac{1}{4} \times \frac{1}{4}-\frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2}\right| \\
&=\left|\frac{3}{16}+\frac{1}{16}-\frac{3}{4}\right|=\left|\frac{3+1-12}{16}\right| \\
&=\left|-\frac{8}{16}\right|=\frac{1}{2} \\
\therefore \quad \theta &=\frac{\pi}{3}
\end{aligned}\)