If `theta` is the angle between two unit vectors `hat(a)` and `hat(b)` then `(1)/(2)|hat(a)-hat(b)|=?`

Options

A. `cos.(theta)/(2)`
B. `sin.(theta)/(2)`
C. `tan.(theta)/(2)`
D. none of these

Explanation

To solve the problem, we need to find the value of \(\frac{1}{2} |\hat{a} - \hat{b}|\), where \(\hat{a}\) and \(\hat{b}\) are unit vectors and \(\theta\) is the angle between them.

### Step-by-Step Solution:

1. **Understanding the Magnitude of the Difference of Vectors**:
We start with the expression \(|\hat{a} - \hat{b}|\). To find this, we can use the property of the dot product:
\[ |\hat{a} - \hat{b}|^2 = (\hat{a} - \hat{b}) \cdot (\hat{a} - \hat{b}) \]

2. **Expanding the Dot Product**:
Expanding the dot product gives us:
\[ |\hat{a} - \hat{b}|^2 = \hat{a} \cdot \hat{a} - 2 \hat{a} \cdot \hat{b} + \hat{b} \cdot \hat{b} \]
Since \(\hat{a}\) and \(\hat{b}\) are unit vectors, we have:
\[ \hat{a} \cdot \hat{a} = 1 \quad \text{and} \quad \hat{b} \cdot \hat{b} = 1 \]
Thus, the equation simplifies to:
\[ |\hat{a} - \hat{b}|^2 = 1 - 2 \hat{a} \cdot \hat{b} + 1 = 2 - 2 \hat{a} \cdot \hat{b} \]

3. **Using the Cosine of the Angle**:
The dot product of two vectors can be expressed in terms of the cosine of the angle between them:
\[ \hat{a} \cdot \hat{b} = \cos(\theta) \]
Therefore, we can substitute this into our equation:
\[ |\hat{a} - \hat{b}|^2 = 2 - 2 \cos(\theta) \]

4. **Factoring the Expression**:
We can factor out the 2:
\[ |\hat{a} - \hat{b}|^2 = 2(1 - \cos(\theta)) \]

5. **Using the Sine Double Angle Identity**:
We know from trigonometric identities that:
\[ 1 - \cos(\theta) = 2 \sin^2\left(\frac{\theta}{2}\right) \]
Substituting this into our equation gives:
\[ |\hat{a} - \hat{b}|^2 = 2 \cdot 2 \sin^2\left(\frac{\theta}{2}\right) = 4 \sin^2\left(\frac{\theta}{2}\right) \]

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

7. **Finding Half the Magnitude**:
Now, we need to find \(\frac{1}{2} |\hat{a} - \hat{b}|\):
\[ \frac{1}{2} |\hat{a} - \hat{b}| = \frac{1}{2} \cdot 2 \sin\left(\frac{\theta}{2}\right) = \sin\left(\frac{\theta}{2}\right) \]

### Final Answer:
\[ \frac{1}{2} |\hat{a} - \hat{b}| = \sin\left(\frac{\theta}{2}\right) \]