Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | Angle between \(\hat{\imath}\) and \(-\hat{\jmath}\) is | (I) | \(\displaystyle\frac{\pi}{6}\) |
| (B) | Angle between \(2\hat{\imath} + \hat{k}\) and \(10\hat{\imath} + 5\hat{k}\) is | (II) | \(\displaystyle\frac{\pi}{4}\) |
| (C) | Angle between \(\hat{\imath}\) and \(\hat{\imath} + \hat{\jmath}\) is | (III) | \(2\pi\) |
| (D) | Angle between \(\sqrt{3}\hat{\jmath} - \hat{k}\) and \(\hat{\jmath}\) is | (IV) | \(\displaystyle\frac{\pi}{2}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(IV), (B)-(III), (C)-(II), (D)-(I)
The angle \(\theta\) between two vectors, with \(0 \le \theta \le \pi\), is given by \(\cos\theta = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}\).
(A) Angle between \(\hat{\imath}\) and \(-\hat{\jmath}\)
This matches (IV) \(\frac{\pi}{2}\): \(\hat{\imath}\cdot(-\hat{\jmath}) = 0\), so the vectors are perpendicular.
(B) Angle between \(2\hat{\imath} + \hat{k}\) and \(10\hat{\imath} + 5\hat{k}\)
This matches (III), by elimination. The printed entry \(2\pi\) does not satisfy it. Since \(10\hat{\imath} + 5\hat{k} = 5(2\hat{\imath} + \hat{k})\), the vectors point in the same direction:
\(\displaystyle \cos\theta = \frac{20 + 5}{\sqrt{5}\cdot 5\sqrt{5}} = 1,\quad \theta = 0\)
(C) Angle between \(\hat{\imath}\) and \(\hat{\imath} + \hat{\jmath}\)
This matches (II) \(\frac{\pi}{4}\):
\(\displaystyle \cos\theta = \frac{1}{1\cdot\sqrt{2}} = \frac{1}{\sqrt{2}},\quad \theta = \frac{\pi}{4}\)
(D) Angle between \(\sqrt{3}\hat{\jmath} - \hat{k}\) and \(\hat{\jmath}\)
This matches (I) \(\frac{\pi}{6}\):
\(\displaystyle \cos\theta = \frac{\sqrt{3}}{\sqrt{3 + 1}\cdot 1} = \frac{\sqrt{3}}{2},\quad \theta = \frac{\pi}{6}\)
So the matching is (A)-(IV), (B)-(III), (C)-(II), (D)-(I).
Note: It seems that there is a mistake in the original NTA paper, which prints (III) as \(2\pi\) when it should read 0. The vectors in (B) point in the same direction, so the angle between them is 0, and an angle between two vectors always lies between 0 and \(\pi\). The question can still be solved because (A), (C) and (D) match only (IV), (II) and (I), which leaves (III) for (B), and Option (3) is the only option with (A)-(IV), (C)-(II), (D)-(I).