Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Direction ratios of two lines | Angle between the lines | ||
| (A) | 4, 3, 1 and 3, −1, 4 | (I) | \(\displaystyle\frac{\pi}{2}\) |
| (B) | 4, 1, 1 and 2, −1, 2 | (II) | \(\displaystyle\frac{\pi}{3}\) |
| (C) | 1, 2, 2 and 2, 1, −2 | (III) | \(\cos^{-1}\left(\displaystyle\frac{8}{9}\right)\) |
| (D) | 1, 2, 2 and 2, 2, 1 | (IV) | \(\displaystyle\frac{\pi}{4}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(II), (B)-(IV), (C)-(I), (D)-(III)
For lines with direction ratios \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\), the angle between the lines is given by
\( \displaystyle \cos\theta = \frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}} \)
(A) 4, 3, 1 and 3, −1, 4
This matches (II) \(\frac{\pi}{3}\):
\( \displaystyle \cos\theta = \frac{12-3+4}{\sqrt{26}\,\sqrt{26}} = \frac{13}{26} = \frac{1}{2} \)
(B) 4, 1, 1 and 2, −1, 2
This matches (IV) \(\frac{\pi}{4}\):
\( \displaystyle \cos\theta = \frac{8-1+2}{\sqrt{18}\,\sqrt{9}} = \frac{9}{9\sqrt{2}} = \frac{1}{\sqrt{2}} \)
(C) 1, 2, 2 and 2, 1, −2
This matches (I) \(\frac{\pi}{2}\):
\((1)(2)+(2)(1)+(2)(-2) = 0\), so \(\cos\theta = 0\)
(D) 1, 2, 2 and 2, 2, 1
This matches (III) \(\cos^{-1}\left(\frac{8}{9}\right)\):
\( \displaystyle \cos\theta = \frac{2+4+2}{3 \times 3} = \frac{8}{9} \)