If `theta` is the angle between the unit vectors `hat a` and `hat b` then `cos (theta/2)` is equal to

Options

A. `|hat a + hat b|`
B. `1/2|hat a - hat b|`
C. `1/2|hat a * hat b|`
D. `1/2|hat a + hat b|`

Answer

To solve the problem of finding \( \cos\left(\frac{\theta}{2}\right) \) where \( \theta \) is the angle between the unit vectors \( \hat{a} \) and \( \hat{b} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that \( \hat{a} \) and \( \hat{b} \) are unit vectors. Therefore, their magnitudes are:
\[ |\hat{a}| = 1 \quad \text{and} \quad |\hat{b}| = 1 \]

2. **Using the Cosine Law**:
The cosine of the angle \( \theta \) between two vectors can be expressed using the dot product:
\[ \hat{a} \cdot \hat{b} = |\hat{a}| |\hat{b}| \cos(\theta) = 1 \cdot 1 \cdot \cos(\theta) = \cos(\theta) \]

3. **Expressing \( \hat{a} + \hat{b} \)**:
We can square the sum of the vectors:
\[ |\hat{a} + \hat{b}|^2 = (\hat{a} + \hat{b}) \cdot (\hat{a} + \hat{b}) = \hat{a} \cdot \hat{a} + \hat{b} \cdot \hat{b} + 2 \hat{a} \cdot \hat{b} \]
Since both \( \hat{a} \) and \( \hat{b} \) are unit vectors:
\[ |\hat{a} + \hat{b}|^2 = 1 + 1 + 2 \cos(\theta) = 2 + 2 \cos(\theta) \]

4. **Using the Cosine Half-Angle Identity**:
We know the identity:
\[ \cos(\theta) = 2 \cos^2\left(\frac{\theta}{2}\right) - 1 \]
Rearranging gives:
\[ \cos(\theta) + 1 = 2 \cos^2\left(\frac{\theta}{2}\right) \]
Therefore:
\[ \cos^2\left(\frac{\theta}{2}\right) = \frac{\cos(\theta) + 1}{2} \]

5. **Substituting Back**:
From the equation \( |\hat{a} + \hat{b}|^2 = 2 + 2 \cos(\theta) \), we can express \( \cos(\theta) \):
\[ \cos(\theta) = \frac{|\hat{a} + \hat{b}|^2 - 2}{2} \]
Substituting this into our half-angle formula:
\[ \cos^2\left(\frac{\theta}{2}\right) = \frac{\left(\frac{|\hat{a} + \hat{b}|^2 - 2}{2}\right) + 1}{2} \]

6. **Final Expression for \( \cos\left(\frac{\theta}{2}\right) \)**:
Simplifying gives:
\[ \cos^2\left(\frac{\theta}{2}\right) = \frac{|\hat{a} + \hat{b}|^2}{4} \]
Taking the square root:
\[ \cos\left(\frac{\theta}{2}\right) = \frac{|\hat{a} + \hat{b}|}{2} \]

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