Question

The angle between the lines whose direction ratios are $1,1,2,\sqrt{3}-1,-\sqrt{3}-1,4$ is.

• A. ${\cos }^{-1}\left(\frac{1}{65}\right)$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (C)

$\frac{\pi }{3}$

We know If $l_1,m_1,n_1$ and $l_2,m_2,n_2$ are direction ratios of the lines, then the angle $\theta$ between them is given by
$\cos \theta =\frac{l_1{l}_2+m_1{m}_2+n_1{n}_2}{\sqrt{l_1^2+m_1^2+n_1^2}\sqrt{l_2^2+m_2^2+n_2^2}}⟶1$
$\therefore$ According to question
$(l_1,m_1,n_1)=(1,1,2)$
$(l_2,m_2,n_2)=(\sqrt{3-1},-\sqrt{3-1},4)$
Using equation 1
$\theta ={\cos }^{-1}\left(\frac{\sqrt{3}-1-\sqrt{3}-1+8}{\sqrt{1^2+1^2+2^2}\sqrt{{(\sqrt{3}-1)}^2+{(\sqrt{3}+1)}^2+4^2}}\right)$
$\theta ={\cos }^{-1}\left(\frac{1}{2}\right)$
$\theta =\frac{\pi }{3}$