If `hat a` and `hatb` are two unit vectors inclined at an angle `theta`, then `sin(theta/2)`

Options

A. `(1)/(2)|a-b|`
B. `(1)/(2)|a+b|`
C. `|a-b|`
D. `|a+b|`

Answer

To solve the problem, we need to find the expression for \(\sin(\theta/2)\) given that \(\hat{a}\) and \(\hat{b}\) are two unit vectors inclined at an angle \(\theta\).

### Step-by-Step Solution:

1. **Understanding the Unit Vectors**:
Since \(\hat{a}\) and \(\hat{b}\) are unit vectors, we have:
\[ |\hat{a}| = 1 \quad \text{and} \quad |\hat{b}| = 1 \]

2. **Using the Formula for Magnitude of the Difference**:
We can use the formula for the magnitude of the difference of two vectors:
\[ |\hat{a} - \hat{b}|^2 = |\hat{a}|^2 + |\hat{b}|^2 - 2|\hat{a}||\hat{b}|\cos(\theta) \]
Substituting the values of the magnitudes:
\[ |\hat{a} - \hat{b}|^2 = 1^2 + 1^2 - 2 \cdot 1 \cdot 1 \cdot \cos(\theta) \]
This simplifies to:
\[ |\hat{a} - \hat{b}|^2 = 2 - 2\cos(\theta) \]

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

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

5. **Taking the Square Root**:
Taking the square root of both sides, we have:
\[ |\hat{a} - \hat{b}| = 2\sin(\theta/2) \]

6. **Solving for \(\sin(\theta/2)\)**:
Dividing both sides by 2, we find:
\[ \sin(\theta/2) = \frac{|\hat{a} - \hat{b}|}{2} \]

### Conclusion:
Thus, the expression for \(\sin(\theta/2)\) in terms of the magnitude of the difference of the unit vectors is:
\[ \sin(\theta/2) = \frac{1}{2} |\hat{a} - \hat{b}| \]