` if veca and vecb ` are unit vectors such that ` veca. vecb = cos theta` , then the value of `| veca + vecb|`, is

Options

A. `2 sin theta//2`
B. ` 2 sin theta`
C. ` 2 cos theta //2`
D. ` 2 cos theta `

Explanation

To solve the problem, we need to find the value of \(|\vec{a} + \vec{b}|\) given that \(\vec{a}\) and \(\vec{b}\) are unit vectors and \(\vec{a} \cdot \vec{b} = \cos \theta\).

### Step-by-Step Solution:

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

2. **Using the Formula for the Magnitude of a Sum of Vectors**:
The magnitude of the sum of two vectors can be calculated using the formula:
\[ |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2(\vec{a} \cdot \vec{b}) \]

3. **Substituting Known Values**:
Substitute the magnitudes and the dot product into the formula:
\[ |\vec{a} + \vec{b}|^2 = 1^2 + 1^2 + 2(\vec{a} \cdot \vec{b}) \]
This simplifies to:
\[ |\vec{a} + \vec{b}|^2 = 1 + 1 + 2(\cos \theta) \]
\[ |\vec{a} + \vec{b}|^2 = 2 + 2\cos \theta \]

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

5. **Using the Trigonometric Identity**:
We know from trigonometric identities that:
\[ 1 + \cos \theta = 2 \cos^2\left(\frac{\theta}{2}\right) \]
Therefore, we can substitute this into our equation:
\[ |\vec{a} + \vec{b}|^2 = 2 \cdot 2 \cos^2\left(\frac{\theta}{2}\right) \]
\[ |\vec{a} + \vec{b}|^2 = 4 \cos^2\left(\frac{\theta}{2}\right) \]

6. **Taking the Square Root**:
Finally, to find \(|\vec{a} + \vec{b}|\), we take the square root of both sides:
\[ |\vec{a} + \vec{b}| = \sqrt{4 \cos^2\left(\frac{\theta}{2}\right)} = 2 \cos\left(\frac{\theta}{2}\right) \]

### Final Answer:
Thus, the value of \(|\vec{a} + \vec{b}|\) is:
\[ |\vec{a} + \vec{b}| = 2 \cos\left(\frac{\theta}{2}\right) \]