Consider the lines \(l_1:\displaystyle\frac{x-1}{0}=\frac{y-1}{1}=\frac{2-z}{1}\) and \(l_2:\displaystyle\frac{x}{2}=\frac{y}{0}=\frac{2z-1}{4}\), then which of the following are correct?
(A) Direction Ratio’s of \(l_1 = \lt 0,1,1 \gt\)
(B) Direction Ratio’s of \(l_2 = \lt 2,0,2 \gt\)
(C) Angle between \(l_1\) and \(l_2 = \displaystyle\frac{\pi}{3}\)
(D) Angle between \(l_1\) and \(l_2 = \displaystyle\frac{2\pi}{3}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (B) and (D) only
In the standard form \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\), the denominators a, b, c are the direction ratios. Since \(\frac{2 - z}{1} = \frac{z - 2}{-1}\) and \(\frac{2z - 1}{4} = \frac{z - \frac{1}{2}}{2}\):
\(\displaystyle l_1:\ \frac{x - 1}{0} = \frac{y - 1}{1} = \frac{z - 2}{-1}\)
\(\displaystyle l_2:\ \frac{x}{2} = \frac{y}{0} = \frac{z - \frac{1}{2}}{2}\)
So the direction ratios are <0, 1, −1> for \(l_1\) and <2, 0, 2> for \(l_2\). With these,
\(\displaystyle \cos\theta = \frac{0(2) + 1(0) + (-1)(2)}{\sqrt{0 + 1 + 1}\,\sqrt{4 + 0 + 4}}\)
\(\displaystyle = \frac{-2}{\sqrt{2}\,\sqrt{8}} = -\frac{1}{2}\)
Two lines make two angles with each other, θ and π − θ, which add up to π. With these direction ratios the angle between \(l_1\) and \(l_2\) is \(\frac{2\pi}{3}\).
So the direction ratios are <0, 1, −1> for \(l_1\) and <2, 0, 2> for \(l_2\), and the angle between the lines is \(\frac{2\pi}{3}\).
Checking the statements
(A) Direction Ratio’s of \(l_1 = \lt 0,1,1 \gt\) — Incorrect. The direction ratios of \(l_1\) are <0, 1, −1>.
(B) Direction Ratio’s of \(l_2 = \lt 2,0,2 \gt\) — Correct. The denominators of \(l_2\) in standard form are 2, 0, 2.
(C) Angle between \(l_1\) and \(l_2 = \frac{\pi}{3}\) — Incorrect. \(\cos\theta\) is \(-\frac{1}{2}\), not \(\frac{1}{2}\).
(D) Angle between \(l_1\) and \(l_2 = \frac{2\pi}{3}\) — Correct. \(\cos\theta = -\frac{1}{2}\) gives \(\theta = \frac{2\pi}{3}\).