Question

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

• A. $\pi$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{4}$

Answer 1

The correct option is C $\frac{\pi }{3}$

Let $(l_1,m_1,n_1)=(\frac{\sqrt{3}}{4},\frac{1}{4},\frac{\sqrt{3}}{2})$ and $(l_2,m_2,n_2)=(\frac{\sqrt{3}}{4},\frac{1}{4},-\frac{\sqrt{3}}{2})$ are the direction cosines of two lines

Angle between two lines with direction cosines $(l_1,m_1,n_1)$ and $(l_2,m_2,n_2)$ is given by

$\cos \theta =|l_1{l}_2+m_1{m}_2+n_1{n}_2|$

$=\left|\begin{array}{c}\frac{\sqrt{3}}{4}\times \frac{\sqrt{3}}{4}+\frac{1}{4}\times \frac{1}{4}+\frac{\sqrt{3}}{2}\times \left(\frac{-\sqrt{3}}{2}\right)\end{array}\right|$
$=\left|\begin{array}{c}\frac{3}{16}+\frac{1}{16}-\frac{3}{4}\end{array}\right|=\left|\begin{array}{c}\frac{-2}{4}\end{array}\right|=\frac{1}{2}$
$\Rightarrow \theta =\frac{\pi }{3}$