The angle between the lines \(l_1:\displaystyle\frac{x+1}{1}=\frac{2-y}{2}=\frac{z-1}{1}\) and \(l_2:\displaystyle\frac{x-1}{4}=\frac{2y-4}{6}=\frac{z-1}{2}\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{\pi}{2}\)
Direction ratios can be read only from the standard form \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\), where x, y and z each have coefficient 1. In this form the y-terms become:
\(\displaystyle l_1:\ \frac{2 - y}{2} = \frac{y - 2}{-2}\), so the direction ratios are 1, −2, 1
\(\displaystyle l_2:\ \frac{2y - 4}{6} = \frac{y - 2}{3}\), so the direction ratios are 4, 3, 2
The angle \(\theta\) between the lines satisfies
\(\displaystyle \cos\theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\,\sqrt{a_2^2 + b_2^2 + c_2^2}}\)
\(a_1a_2 + b_1b_2 + c_1c_2 = (1)(4) + (-2)(3) + (1)(2) = 0\)
So \(\cos\theta = 0\) and the lines are perpendicular: \(\theta = \frac{\pi}{2}\).