Let `vec a= hat i` be a vector which makes an angle of `120^@` with a unit vector`vec b` in XY plane. then the unit vector `(vec a+ vec b)` is

Options

A. `-(1)/(2)hati+(sqrt(3))/(2)hatj`
B. `-(sqrt(3))/(2)hati+(1)/(2)hatj`
C. `(1)/(2)hati+(sqrt(3))/(2)hatj`
D. `(sqrt(3))/(2)hati-(1)/(2)hatj`

Explanation

To solve the problem, we need to find the unit vector \((\vec{a} + \vec{b})\) given that \(\vec{a} = \hat{i}\) and it makes an angle of \(120^\circ\) with the unit vector \(\vec{b}\) in the XY plane.

### Step-by-Step Solution:

1. **Identify the Given Vectors**:
- We have \(\vec{a} = \hat{i}\).
- The angle between \(\vec{a}\) and \(\vec{b}\) is \(120^\circ\).

2. **Express the Unit Vector \(\vec{b}\)**:
- Since \(\vec{b}\) is a unit vector, we can express it in terms of its components using the angle it makes with \(\vec{a}\).
- The components of \(\vec{b}\) can be expressed as:
\[ \vec{b} = \cos(120^\circ) \hat{i} + \sin(120^\circ) \hat{j} \]

3. **Calculate \(\cos(120^\circ)\) and \(\sin(120^\circ)\)**:
- We know:
\[ \cos(120^\circ) = -\frac{1}{2}, \quad \sin(120^\circ) = \frac{\sqrt{3}}{2} \]
- Therefore, substituting these values into the expression for \(\vec{b}\):
\[ \vec{b} = -\frac{1}{2} \hat{i} + \frac{\sqrt{3}}{2} \hat{j} \]

4. **Add the Vectors \(\vec{a}\) and \(\vec{b}\)**:
- Now we can find \(\vec{a} + \vec{b}\):
\[ \vec{a} + \vec{b} = \hat{i} + \left(-\frac{1}{2} \hat{i} + \frac{\sqrt{3}}{2} \hat{j}\right) \]
- Simplifying this gives:
\[ \vec{a} + \vec{b} = \left(1 - \frac{1}{2}\right) \hat{i} + \frac{\sqrt{3}}{2} \hat{j} = \frac{1}{2} \hat{i} + \frac{\sqrt{3}}{2} \hat{j} \]

5. **Find the Magnitude of \(\vec{a} + \vec{b}\)**:
- The magnitude of the vector \(\vec{a} + \vec{b}\) is:
\[ |\vec{a} + \vec{b}| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

6. **Determine the Unit Vector**:
- Since the magnitude of \(\vec{a} + \vec{b}\) is 1, the unit vector \(\hat{u}\) in the direction of \(\vec{a} + \vec{b}\) is:
\[ \hat{u} = \frac{\vec{a} + \vec{b}}{|\vec{a} + \vec{b}|} = \vec{a} + \vec{b} \]
- Thus, the unit vector \((\vec{a} + \vec{b})\) is:
\[ \hat{u} = \frac{1}{2} \hat{i} + \frac{\sqrt{3}}{2} \hat{j} \]

### Final Answer:
The unit vector \((\vec{a} + \vec{b})\) is:
\[ \hat{u} = \frac{1}{2} \hat{i} + \frac{\sqrt{3}}{2} \hat{j} \]