Mathematics In Physics Question 7
Question 7 - 2024 (31 Jan Shift 2)
If two vectors $\vec{A}$ and $\vec{B}$ having equal magnitude $\mathrm{R}$ are inclined at angle $\theta$, then
(1) $|\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}|=\sqrt{2} \mathrm{R} \sin \left(\frac{\theta}{2}\right)$
(2) $|\vec{A}+\vec{B}|=2 R \sin \left(\frac{\theta}{2}\right)$
(3) $|\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}|=2 \mathrm{R} \cos \left(\frac{\theta}{2}\right)$
(4) $|\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}|=2 R \cos \left(\frac{\theta}{2}\right)$
Show Answer
Answer: (3)
Solution:
The magnitude of resultant vector
$R^{\prime}=\sqrt{a^{2}+b^{2}+2 a b \cos \theta}$
Here $\mathrm{a}=\mathrm{b}=\mathrm{R}$
Then $R^{\prime}=\sqrt{R^{2}+R^{2}+2 R^{2} \cos \theta}$
$=R \sqrt{2} \sqrt{1+\cos \theta}$
$=\sqrt{2} R \sqrt{2 \cos ^{2} \frac{\theta}{2}}$
$=2 R \cos \frac{\theta}{2}$
