Cosine of the acute angle between the lines \(\displaystyle\frac{x-3}{2}=\frac{y-2}{1}=\frac{z-5}{2}\) and \(\displaystyle\frac{x-1}{6}=\frac{y-3}{-3}=\frac{z+6}{2}\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{13}{21}\)
The direction ratios are the denominators: (2, 1, 2) and (6, −3, 2). For the acute angle between the lines:
\( \displaystyle \cos\theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\,\sqrt{a_2^2 + b_2^2 + c_2^2}} \)
\(a_1a_2 + b_1b_2 + c_1c_2 = 12 - 3 + 4 = 13\)
\(\sqrt{4 + 1 + 4} = 3,\quad \sqrt{36 + 9 + 4} = 7\)
\( \displaystyle \cos\theta = \frac{13}{3 \times 7} = \frac{13}{21} \)