Question

If \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) are unit vectors and \(\theta\) is the angle between them, then \(\tan \frac{\theta}{2}=\)

• A. \(|\overset{-}{\mathrm{a}}-\overset{-}{\mathrm{b}}|\)
• B. \(|\vec{a}+\vec{b}|\)
• C. \(\frac{|\vec{a}+\vec{b}|}{|\vec{a}-\vec{b}|}\)
• D. \(\frac{|\overline{a}-\overline{b}|}{|\overline{a}+\overline{b}|}\)

Hint:

For unit vectors, \(|\overline{a}\pm\overline{b}|^2 = 2 \pm 2\cos\theta\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
For unit vectors \(|\overline{a}|=|\overline{b}|=1\), so \(|\overline{a}+\overline{b}|^2 = 2 + 2\cos\theta\) and \(|\overline{a}-\overline{b}|^2 = 2 - 2\cos\theta\).

Step 2: Half angle form
\(|\overline{a}+\overline{b}|^2 = 4\cos^2\dfrac{\theta}{2}\) and \(|\overline{a}-\overline{b}|^2 = 4\sin^2\dfrac{\theta}{2}\).

Step 3: Take the ratio
\(\tan\dfrac{\theta}{2} = \dfrac{\sin(\theta/2)}{\cos(\theta/2)} = \dfrac{|\overline{a}-\overline{b}|}{|\overline{a}+\overline{b}|}\)

Final Answer:
\(\tan\dfrac{\theta}{2} = \dfrac{|\overline{a}-\overline{b}|}{|\overline{a}+\overline{b}|}\). \[ \boxed{\text{(D) }\dfrac{|\overline{a}-\overline{b}|}{|\overline{a}+\overline{b}|}} \]