Q68 / 150 · MHT CET 2021 (20 Sep Shift 1) · Mathematics
The angle between a line with direction ratios 2, 2, 1 and a line joining $(3,1,4)$ and $(7,2,12)$ is
- A$\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right)$
- B$\cos ^{-1}\left(\frac{1}{3}\right)$
- Correct answerC$\cos ^{-1}\left(\frac{2}{3}\right)$
- D$\cos ^{-1}\left(\frac{\sqrt{2}}{3}\right)$
Solution
Direction ratios of a line joining $(3,1,4)$ and $(7,2,12)$ are $4,1,8$
Let $\left(a_1, b_1, c_1\right)=(2,2,1)$ and $\left(a_2, c_2\right)=(4,1,8)$.
Hence angle $\theta$ between the lines is given by
$$
\begin{aligned}
& \cos \theta=\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1 \sqrt{a_2^2+b_2^2+c_2^2}}} \\
& =\frac{8+2+8}{\sqrt{4+4+1} \cdot \sqrt{16+1+64}}=\frac{18}{\sqrt{9} \cdot \sqrt{81}}=\frac{18}{(3)(9)}=\frac{2}{3} \\
& \therefore \theta=\cos ^{-1}\left(\frac{2}{3}\right)
\end{aligned}
$$
Hence, the correct answer is option: 'C'.