Let `hat(a), hat(b) ` be two unit vectors and `theta` be the angle between them.
What is `cos ((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 find \( \cos\left(\frac{\theta}{2}\right) \) where \( \hat{a} \) and \( \hat{b} \) are unit vectors and \( \theta \) is the angle between them, we can follow these steps:

### Step 1: Understand the properties of unit vectors
Since \( \hat{a} \) and \( \hat{b} \) are unit vectors, we have:
\[ |\hat{a}| = 1 \quad \text{and} \quad |\hat{b}| = 1 \]

### Step 2: Use the formula for the cosine of the angle between two vectors
The cosine of the angle \( \theta \) between the two vectors can be expressed as:
\[ \cos(\theta) = \hat{a} \cdot \hat{b} \]

### Step 3: Calculate the magnitude of \( \hat{a} + \hat{b} \)
To find \( \cos\left(\frac{\theta}{2}\right) \), we first need to calculate the magnitude of \( \hat{a} + \hat{b} \):
\[ |\hat{a} + \hat{b}|^2 = (\hat{a} + \hat{b}) \cdot (\hat{a} + \hat{b}) \]
Expanding this using the dot product:
\[ |\hat{a} + \hat{b}|^2 = \hat{a} \cdot \hat{a} + 2 \hat{a} \cdot \hat{b} + \hat{b} \cdot \hat{b} \]
Since \( \hat{a} \cdot \hat{a} = 1 \) and \( \hat{b} \cdot \hat{b} = 1 \):
\[ |\hat{a} + \hat{b}|^2 = 1 + 2\cos(\theta) + 1 = 2 + 2\cos(\theta) \]

### Step 4: Relate \( \cos(\theta) \) to \( \cos\left(\frac{\theta}{2}\right) \)
Using the half-angle identity:
\[ \cos(\theta) = 2\cos^2\left(\frac{\theta}{2}\right) - 1 \]
We can rearrange this to express \( \cos^2\left(\frac{\theta}{2}\right) \):
\[ \cos(\theta) + 1 = 2\cos^2\left(\frac{\theta}{2}\right) \]
Thus,
\[ \cos^2\left(\frac{\theta}{2}\right) = \frac{\cos(\theta) + 1}{2} \]

### Step 5: Substitute \( \cos(\theta) \) into the equation
From our earlier calculation, we have:
\[ \cos(\theta) = \frac{|\hat{a} + \hat{b}|^2 - 2}{2} \]
Substituting this into the equation for \( \cos^2\left(\frac{\theta}{2}\right) \):
\[ \cos^2\left(\frac{\theta}{2}\right) = \frac{\left(\frac{|\hat{a} + \hat{b}|^2 - 2}{2}\right) + 1}{2} \]
This simplifies to:
\[ \cos^2\left(\frac{\theta}{2}\right) = \frac{|\hat{a} + \hat{b}|^2}{4} \]

### Step 6: Take the square root to find \( \cos\left(\frac{\theta}{2}\right) \)
Taking the square root gives us:
\[ \cos\left(\frac{\theta}{2}\right) = \frac{|\hat{a} + \hat{b}|}{2} \]

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