Let `hat(a), hat(b) ` be two unit vectors and `theta` be the angle between them.
What is `sin ((theta)/(2))` equal to ?

Options

A. `(|hat(a)-hat(b)|)/(2)`
B. `(|hat(a)+hat(b)|)/(2)`
C. `(|hat(i)-hat(b)|)/(4)`
D. `(|hat(a)+hat(b)|)/(4)`

Explanation

To solve the problem, we need to find the value of \( \sin\left(\frac{\theta}{2}\right) \) where \( \theta \) is the angle between two unit vectors \( \hat{a} \) and \( \hat{b} \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two unit vectors \( \hat{a} \) and \( \hat{b} \) and the angle between them is \( \theta \). We need to express \( \sin\left(\frac{\theta}{2}\right) \) in terms of the vectors.

2. **Using the Dot Product**:
The dot product of two vectors \( \hat{a} \) and \( \hat{b} \) can be expressed as:
\[ \hat{a} \cdot \hat{b} = |\hat{a}| |\hat{b}| \cos(\theta) \]
Since both vectors are unit vectors, we have:
\[ \hat{a} \cdot \hat{b} = 1 \cdot 1 \cdot \cos(\theta) = \cos(\theta) \]

3. **Finding the Magnitude of \( \hat{a} - \hat{b} \)**:
We can find the magnitude of the vector \( \hat{a} - \hat{b} \):
\[ |\hat{a} - \hat{b}|^2 = (\hat{a} - \hat{b}) \cdot (\hat{a} - \hat{b}) \]
Expanding this, we get:
\[ |\hat{a}|^2 - 2(\hat{a} \cdot \hat{b}) + |\hat{b}|^2 \]
Since \( |\hat{a}|^2 = 1 \) and \( |\hat{b}|^2 = 1 \), we have:
\[ |\hat{a} - \hat{b}|^2 = 1 - 2\cos(\theta) + 1 = 2 - 2\cos(\theta) = 2(1 - \cos(\theta)) \]

4. **Using the Half-Angle Identity**:
We know from trigonometric identities that:
\[ 1 - \cos(\theta) = 2\sin^2\left(\frac{\theta}{2}\right) \]
Therefore, we can substitute this into our expression for \( |\hat{a} - \hat{b}|^2 \):
\[ |\hat{a} - \hat{b}|^2 = 2(2\sin^2\left(\frac{\theta}{2}\right)) = 4\sin^2\left(\frac{\theta}{2}\right) \]

5. **Taking the Square Root**:
Taking the square root of both sides gives us:
\[ |\hat{a} - \hat{b}| = 2\sin\left(\frac{\theta}{2}\right) \]

6. **Solving for \( \sin\left(\frac{\theta}{2}\right) \)**:
Rearranging the equation, we find:
\[ \sin\left(\frac{\theta}{2}\right) = \frac{|\hat{a} - \hat{b}|}{2} \]

### Final Result:
Thus, we conclude that:
\[ \sin\left(\frac{\theta}{2}\right) = \frac{|\hat{a} - \hat{b}|}{2} \]