If `vec(a) and vec(b)` are unit vectors inclined at an angle of `30^(@)` to each other, then which one of the following is correct?

Options

A. `|vec(a)+vec(b)|gt1`
B. `1 gt |vec(a)+vec(b)|lt2`
C. `|vec(a)+vec(b)|=2`
D. `|vec(a)+vec(b)|lt2`

Explanation

To solve the problem, we need to find the magnitude of the sum of two unit vectors \( \vec{a} \) and \( \vec{b} \) that are inclined at an angle of \( 30^\circ \) to each other.

### Step-by-Step Solution:

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

2. **Using the Cosine Rule**:
The magnitude of the sum of two vectors can be calculated using the formula:
\[ |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2 |\vec{a}| |\vec{b}| \cos \theta \]
where \( \theta \) is the angle between the two vectors.

3. **Substituting Values**:
Here, \( |\vec{a}| = 1 \), \( |\vec{b}| = 1 \), and \( \theta = 30^\circ \). Substituting these values into the formula gives:
\[ |\vec{a} + \vec{b}|^2 = 1^2 + 1^2 + 2 \cdot 1 \cdot 1 \cdot \cos(30^\circ) \]

4. **Calculating \( \cos(30^\circ) \)**:
We know that:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
Therefore, substituting this into the equation:
\[ |\vec{a} + \vec{b}|^2 = 1 + 1 + 2 \cdot \frac{\sqrt{3}}{2} \]

5. **Simplifying the Expression**:
This simplifies to:
\[ |\vec{a} + \vec{b}|^2 = 2 + \sqrt{3} \]

6. **Finding the Magnitude**:
To find \( |\vec{a} + \vec{b}| \), we take the square root:
\[ |\vec{a} + \vec{b}| = \sqrt{2 + \sqrt{3}} \]

7. **Estimating the Value**:
We can estimate \( \sqrt{3} \approx 1.732 \), so:
\[ 2 + \sqrt{3} \approx 2 + 1.732 = 3.732 \]
Therefore,
\[ |\vec{a} + \vec{b}| \approx \sqrt{3.732} \approx 1.93 \]

8. **Conclusion**:
Since \( 1 < |\vec{a} + \vec{b}| < 2 \), we conclude that the magnitude of \( \vec{a} + \vec{b} \) lies between 1 and 2.

### Final Answer:
The correct option is that the magnitude of \( \vec{a} + \vec{b} \) is greater than 1 and less than 2.