Acute angle between the lines \(\displaystyle\frac{x}{3}=\frac{y}{4}=\frac{z}{5}\) and \(\displaystyle\frac{x-1}{4}=\frac{y+1}{-3}=\frac{z+10}{5}\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{\pi}{3}\)
For lines with direction ratios \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\), the acute angle \(\theta\) between them is given by
\(\displaystyle \cos\theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\,\sqrt{a_2^2 + b_2^2 + c_2^2}}\)
The denominators give the direction ratios 3, 4, 5 and 4, −3, 5.
\(a_1a_2 + b_1b_2 + c_1c_2 = 12 - 12 + 25 = 25\)
\(\sqrt{9 + 16 + 25} = \sqrt{50}\), \(\sqrt{16 + 9 + 25} = \sqrt{50}\)
\(\displaystyle \cos\theta = \frac{25}{\sqrt{50}\,\sqrt{50}}\)
\(\displaystyle = \frac{25}{50} = \frac{1}{2}\)
\(\displaystyle \theta = \frac{\pi}{3}\)