If $\hat{a}$ and $\hat{b}$ are two unit vectors and $ heta$ is angle between them, then $|\frac{\hat{a}-\hat{b}}{2}|$ is equal to:

Q: If $\hat{a}$ and $\hat{b}$ are two unit vectors and $\theta$ is angle between them, then $|\frac{\hat{a}-\hat{b}}{2}|$ is equal to:

If $\hat{a}$ and $\hat{b}$ are two unit vectors and $\theta$ is angle between them, then $|\frac{\hat{a}-\hat{b}}{2}|$ is equal to:

Question

If $\hat{a}$ and $\hat{b}$ are two unit vectors and $	heta$ is angle between them, then $|\frac{\hat{a}-\hat{b}}{2}|$ is equal to:

Accepted Answer

To determine the value of $ \left|\frac{\hat{a} - \hat{b}}{2}\right| $ when $\hat{a}$ and $\hat{b}$ are unit vectors and $\theta$ is the angle between them, we start by considering the properties of vectors and trigonometric identities.

1. **Magnitude of the Difference of Two Vectors:**

The magnitude of the difference between two vectors $\hat{a}$ and $\hat{b}$ is given by:
   $$
   |\hat{a} - \hat{b}| = \sqrt{(\hat{a} - \hat{b}) \cdot (\hat{a} - \hat{b})}
   $$
   Expanding the dot product, we have:
   $$
   (\hat{a} - \hat{b}) \cdot (\hat{a} - \hat{b}) = \hat{a} \cdot \hat{a} - 2(\hat{a} \cdot \hat{b}) + \hat{b} \cdot \hat{b}
   $$
   Since $\hat{a}$ and $\hat{b}$ are unit vectors, $\hat{a} \cdot \hat{a} = 1$ and $\hat{b} \cdot \hat{b} = 1$. Therefore:
   $$
   |\hat{a} - \hat{b}| = \sqrt{1 - 2(\hat{a} \cdot \hat{b}) + 1} = \sqrt{2 - 2(\hat{a} \cdot \hat{b})}
   $$

2. **Relation to the Angle $\theta$:**

The dot product $\hat{a} \cdot \hat{b}$ can be expressed in terms of the angle $\theta$ as:
   $$
   \hat{a} \cdot \hat{b} = \cos \theta
   $$
   Substituting this into the expression for the magnitude:
   $$
   |\hat{a} - \hat{b}| = \sqrt{2 - 2\cos \theta} = \sqrt{2(1 - \cos \theta)}
   $$

3. **Using the Trigonometric Identity:**

The identity for the sine of half an angle is:
   $$
   1 - \cos \theta = 2 \sin^2\left(\frac{\theta}{2}\right)
   $$
   Substituting this into the expression:
   $$
   |\hat{a} - \hat{b}| = \sqrt{2 \cdot 2 \sin^2\left(\frac{\theta}{2}\right)} = 2 \sin\left(\frac{\theta}{2}\right)
   $$

4. **Final Calculation:**

Therefore, the expression for $\left|\frac{\hat{a} - \hat{b}}{2}\right|$ is:
   $$
   \left|\frac{\hat{a} - \hat{b}}{2}\right| = \frac{1}{2} \times 2 \sin\left(\frac{\theta}{2}\right) = \sin\left(\frac{\theta}{2}\right)
   $$

Thus, the correct answer is Option 1: $ \sin\left(\frac{\theta}{2}\right) $.

**Explanation for Incorrect Options:**

- **Option 2: $ \sin \theta $** is incorrect because the expression derived involves the sine of half the angle, not the full angle.
- **Option 3: $ 2 \sin \theta $** is incorrect because the factor of 2 is not applicable to the full angle sine in this context.
- **Option 4: $ \sin 2\theta $** is incorrect because the expression involves the sine of half the angle, not double the angle.

Therefore, the correct answer is indeed Option 1: $ \sin\left(\frac{\theta}{2}\right) $.