Let `hat(a), hat(b)` be two unit vectors and 0 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(a)- hat(b)|)/(4)`
D. `(|hat(a) + hat(b)|)/(4)`

Answer

To solve the problem of finding \( \cos\left(\frac{\theta}{2}\right) \) 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 relationship between the vectors
Since \( \hat{a} \) and \( \hat{b} \) are unit vectors, we have:
\[ |\hat{a}| = 1 \quad \text{and} \quad |\hat{b}| = 1 \]
The cosine of the angle \( \theta \) between them can be expressed using the dot product:
\[ \hat{a} \cdot \hat{b} = |\hat{a}| |\hat{b}| \cos(\theta) = \cos(\theta) \]

### Step 2: Use the formula for the sum of vectors
We can find the magnitude of the sum of the vectors:
\[ |\hat{a} + \hat{b}|^2 = |\hat{a}|^2 + |\hat{b}|^2 + 2 \hat{a} \cdot \hat{b} \]
Substituting the values:
\[ |\hat{a} + \hat{b}|^2 = 1^2 + 1^2 + 2 \cos(\theta) = 2 + 2 \cos(\theta) \]

### Step 3: Relate the result to \( \cos\left(\frac{\theta}{2}\right) \)
We know from trigonometric identities that:
\[ 1 + \cos(\theta) = 2 \cos^2\left(\frac{\theta}{2}\right) \]
Thus, we can rewrite our equation:
\[ |\hat{a} + \hat{b}|^2 = 2 + 2(2 \cos^2\left(\frac{\theta}{2}\right) - 1) \]
This simplifies to:
\[ |\hat{a} + \hat{b}|^2 = 4 \cos^2\left(\frac{\theta}{2}\right) \]

### Step 4: Solve for \( \cos\left(\frac{\theta}{2}\right) \)
Taking the square root of both sides gives:
\[ |\hat{a} + \hat{b}| = 2 \cos\left(\frac{\theta}{2}\right) \]
Thus, we can express \( \cos\left(\frac{\theta}{2}\right) \) as:
\[ \cos\left(\frac{\theta}{2}\right) = \frac{|\hat{a} + \hat{b}|}{2} \]

### Conclusion
The final expression for \( \cos\left(\frac{\theta}{2}\right) \) is:
\[ \cos\left(\frac{\theta}{2}\right) = \frac{|\hat{a} + \hat{b}|}{2} \]