The angle between the lines whose direction c | vedclass

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(C) Given the direction cosines of the two lines as $(l_1, m_1, n_1) = \left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\right)$ and $(l_2, m_

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$\frac{\pi}{3}$

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(C) Given the direction cosines of the two lines as $(l_1, m_1, n_1) = \left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}ight)$ and $(l_2, m_2, n_2) = \left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2}ight)$.The cosine of the angle $	heta$ between two lines with direction cosines $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ is given by $\cos 	heta = |l_1 l_2 + m_1 m_2 + n_1 n_2|$.Substituting the values:$\cos 	heta = \left| \left(\frac{\sqrt{3}}{4}ight)\left(\frac{\sqrt{3}}{4}ight) + \left(\frac{1}{4}ight)\left(\frac{1}{4}ight) + \left(\frac{\sqrt{3}}{2}ight)\left(-\frac{\sqrt{3}}{2}ight) ight|$$\cos 	heta = \left| \frac{3}{16} + \frac{1}{16} - \frac{3}{4} ight|$$\cos 	heta = \left| \frac{4}{16} - \frac{12}{16} ight| = \left| -\frac{8}{16} ight| = \left| -\frac{1}{2} ight| = \frac{1}{2}$.Since $\cos 	heta = \frac{1}{2}$,we have $	heta = \frac{\pi}{3}$.

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:

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