If \(|\vec{A}|=2\) and \(|\vec{B}|=4\), then match the relations in Column I with the angle \(\theta\) between \(\vec{A}\) and \(\vec{B}\) in Column II.
| Column I | Column II | ||
| (a) | \(\vec{A}.\vec{B}=0\) | (i) | \(\theta=0^{\circ}\) |
| (b) | \(\vec{A}.\vec{B}=8\) | (ii) | \(\theta=90^{\circ}\) |
| (c) | \(\vec{A}.\vec{B}=4\) | (iii) | \(\theta=180^{\circ}\) |
| (d) | \(\vec{A}.\vec{B}=-8\) | (iv) | \(\theta=60^{\circ}\) |
Choose the correct answer from the options given below:
| 1. | (a)–(iii), (b)-(ii), (c)-(i), (d)-(iv) |
| 2. | (a)–(ii), (b)-(i), (c)-(iv), (d)-(iii) |
| 3. | (a)–(ii), (b)-(iv), (c)-(iii), (d)-(i) |
| 4. | (a)–(iii), (b)-(i), (c)-(ii), (d)-(iv) |
Step: Analyse all the options.
\(\text{Given: $|A|=2$ and $|B|=4$}\\ (a) ~\vec A.\vec B=A B \cos \theta=0 \\ \Rightarrow \quad 2 \times 4 \cos \theta=0 \\ \Rightarrow \quad \cos \theta=0=\cos 90^{\circ} \\ \Rightarrow \quad \theta=90^{\circ} \\\text{Therefore, option (a) matches with option (ii)}. \)
\((b) ~\vec A. \vec B=AB \cos \theta=8 \\ \\ \Rightarrow \quad 2 \times 4 \cos \theta=8 \\\Rightarrow \quad \cos \theta=1=\cos0^{\circ}\Rightarrow \theta= 0^{\circ} \\\text{Therefore, option (b) matches with option (i)}.\)
\( (c) ~ \vec A.\vec B= A B \cos \theta=4 \\\Rightarrow 2 \times 4 \cos \theta=4 \\ \Rightarrow \cos \theta=\frac{1}{2}=\cos 60^{\circ} \Rightarrow \theta=60^{\circ} \\ \text{Therefore, option (c) matches with option (iv)}.\)
\( (d) ~ \vec A.\vec B=AB \cos \theta=-8 \\ \Rightarrow 2 \times 4 \cos \theta=-8 \\ \Rightarrow \cos \theta=-1=\cos 180^{\circ} \\\Rightarrow \theta=180^{\circ} \\ \text{Therefore, option (d) matches with option (iii).} \)
Therefore, the correct match is \(\mathrm{(a)}\rightarrow(\mathrm{ii}); \mathrm{(b)}\rightarrow(\mathrm{i});\mathrm{(c)}\rightarrow(\mathrm{iv});\mathrm{(d)}\rightarrow(\mathrm{iii})\)
Hence, option (2) is the correct answer.
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