Question

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

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

Answer 1

Answer: (C)

$\frac{2\pi }{3}$

The direction ratios of two lines is given $\left(\frac{\sqrt{3}}{4},\frac{1}{4},\frac{\sqrt{3}}{2}\right)$ and $(\frac{\sqrt{3}}{4},\frac{1}{4},\frac{-\sqrt{3}}{2})$

Angle between two lines is given by

$\cos \theta =(\frac{\sqrt{3}}{4},\frac{1}{4},\frac{\sqrt{3}}{2})\cdot (\frac{\sqrt{3}}{4},\frac{1}{4},\frac{-\sqrt{3}}{2})$

$=\frac{\sqrt{3}}{4}\times \frac{\sqrt{3}}{4}+\frac{1}{4}\times \frac{1}{4}-\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{2}$

$=\frac{3}{16}+\frac{1}{16}-\frac{3}{4}=-\frac{1}{2}$

$⟹\cos \theta =-\frac{1}{2}$
$⟹\theta =120\text{or}240$

$\therefore \cos \theta =\frac{2\pi }{3}$