Match List-I with List-II
| List-I (vector) | List-II (angle with the positive x-axis) | ||
| (A) | \(\hat{\imath}+\sqrt{3}\,\hat{\jmath}\) | (I) | \(\displaystyle\frac{\pi}{2}\) |
| (B) | \(\hat{\imath}-\hat{\jmath}\) | (II) | \(\displaystyle\frac{\pi}{3}\) |
| (C) | \(-\hat{\imath}+\sqrt{3}\,\hat{k}\) | (III) | \(\displaystyle\frac{2\pi}{3}\) |
| (D) | \(\hat{\jmath}+\hat{k}\) | (IV) | \(\displaystyle\frac{\pi}{4}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(II), (B)-(IV), (C)-(III), (D)-(I)
The angle \(\alpha\) with the positive x-axis has \(\cos\alpha = \displaystyle\frac{\vec{v}\cdot\hat{\imath}}{|\vec{v}|}\), the first direction cosine of \(\vec{v}\), with \(0 \le \alpha \le \pi\).
(A) \(\hat{\imath}+\sqrt{3}\,\hat{\jmath}\)
This matches (II) \(\frac{\pi}{3}\).
\( \displaystyle \cos\alpha = \frac{1}{\sqrt{1+3}} = \frac{1}{2} \)
(B) \(\hat{\imath}-\hat{\jmath}\)
This matches (IV) \(\frac{\pi}{4}\).
\( \displaystyle \cos\alpha = \frac{1}{\sqrt{2}} \)
(C) \(-\hat{\imath}+\sqrt{3}\,\hat{k}\)
This matches (III) \(\frac{2\pi}{3}\).
\( \displaystyle \cos\alpha = \frac{-1}{\sqrt{1+3}} = -\frac{1}{2} \)
(D) \(\hat{\jmath}+\hat{k}\)
This matches (I) \(\frac{\pi}{2}\).
\(\cos\alpha = 0\)
The vector has no x-component, so it is perpendicular to the x-axis.