Q81 / 180 · COMEDK 2024 (Memory Based) · Mathematics
The angle between the lines $\frac{x-2}{3}=\frac{y-1}{1}=\frac{z+3}{-1}$ and $\frac{x+1}{2}=\frac{y-1}{-1}=\frac{z}{2}$ is
- A$\cos ^{-1}\left(\frac{3}{\sqrt{11}}\right)$
- B$\sin ^{-1}\left(\frac{3}{\sqrt{11}}\right)$
- Correct answerC$\cos ^{-1}\left(\frac{1}{\sqrt{11}}\right)$
- D$\sin ^{-1}\left(\frac{1}{\sqrt{11}}\right)$
Solution
Let, $L_1 \equiv \frac{x-2}{3}=\frac{y-1}{1}=\frac{z+3}{-1}$
$\therefore$ Direction ratio of $L_1\left(l_1, m_1, n_1\right)=(3,1,-1)$
$L_2 \equiv \frac{x+1}{2}=\frac{y-1}{-1}=\frac{z-0}{2}$
$\therefore$ Direction ratio of $L_2\left(l_2, m_2, n_2\right)=(2,-1,2)$
$\begin{aligned}<br/>& \text { Angle between } L_1 \text { and } L_2 \\ <br/>& =\cos \theta=\frac{l_1 L_2+m_1 m_2+n_1 n_2}{\sqrt{l_1^2+m_1^2+n_1^2} \sqrt{l_2^2+m_2^2+n_2^2}} \\<br/>& \cos \theta=\frac{6-1-2}{\sqrt{9+1+1} \sqrt{4+1+4}}=\frac{3}{3 \sqrt{11}}=\frac{1}{\sqrt{11}} \\<br/>& \theta=\cos ^{-1}\left(\frac{1}{\sqrt{11}}\right)<br/>\end{aligned}$
$\therefore$ Direction ratio of $L_1\left(l_1, m_1, n_1\right)=(3,1,-1)$
$L_2 \equiv \frac{x+1}{2}=\frac{y-1}{-1}=\frac{z-0}{2}$
$\therefore$ Direction ratio of $L_2\left(l_2, m_2, n_2\right)=(2,-1,2)$
$\begin{aligned}<br/>& \text { Angle between } L_1 \text { and } L_2 \\ <br/>& =\cos \theta=\frac{l_1 L_2+m_1 m_2+n_1 n_2}{\sqrt{l_1^2+m_1^2+n_1^2} \sqrt{l_2^2+m_2^2+n_2^2}} \\<br/>& \cos \theta=\frac{6-1-2}{\sqrt{9+1+1} \sqrt{4+1+4}}=\frac{3}{3 \sqrt{11}}=\frac{1}{\sqrt{11}} \\<br/>& \theta=\cos ^{-1}\left(\frac{1}{\sqrt{11}}\right)<br/>\end{aligned}$
Hence, the correct answer is option: 'C'.