If `hat(e_1), hat(e_2)` and `hat(e_1)+hat(e_2)` are unit vectors, then angle between `hat(e_1) and hat(e_2)`is

Options

A. `90^(@)`
B. `120^(@)`
C. `450^(@)`
D. `135^(@)`

Answer

To solve the problem, we need to find the angle between the unit vectors \(\hat{e_1}\) and \(\hat{e_2}\) given that both are unit vectors and their sum \(\hat{e_1} + \hat{e_2}\) is also a unit vector.

### Step-by-step Solution:

1. **Understanding Unit Vectors**:
Since \(\hat{e_1}\) and \(\hat{e_2}\) are unit vectors, we have:
\[ |\hat{e_1}| = 1 \quad \text{and} \quad |\hat{e_2}| = 1 \]

2. **Using the Property of the Sum of Vectors**:
The sum of the vectors \(\hat{e_1} + \hat{e_2}\) is also a unit vector:
\[ |\hat{e_1} + \hat{e_2}| = 1 \]

3. **Squaring the Magnitude of the Sum**:
We square both sides:
\[ |\hat{e_1} + \hat{e_2}|^2 = 1^2 \]
Expanding the left-hand side using the dot product:
\[ (\hat{e_1} + \hat{e_2}) \cdot (\hat{e_1} + \hat{e_2}) = \hat{e_1} \cdot \hat{e_1} + 2 \hat{e_1} \cdot \hat{e_2} + \hat{e_2} \cdot \hat{e_2} \]

4. **Substituting the Values**:
Since \(|\hat{e_1}|^2 = 1\) and \(|\hat{e_2}|^2 = 1\):
\[ 1 + 2 \hat{e_1} \cdot \hat{e_2} + 1 = 1 \]
This simplifies to:
\[ 2 + 2 \hat{e_1} \cdot \hat{e_2} = 1 \]

5. **Isolating the Dot Product**:
Rearranging gives:
\[ 2 \hat{e_1} \cdot \hat{e_2} = 1 - 2 \]
\[ 2 \hat{e_1} \cdot \hat{e_2} = -1 \]
Thus:
\[ \hat{e_1} \cdot \hat{e_2} = -\frac{1}{2} \]

6. **Relating Dot Product to Angle**:
The dot product of two vectors can be expressed in terms of the angle \(\theta\) between them:
\[ \hat{e_1} \cdot \hat{e_2} = |\hat{e_1}| |\hat{e_2}| \cos \theta \]
Since both are unit vectors, this simplifies to:
\[ \hat{e_1} \cdot \hat{e_2} = \cos \theta \]
Therefore:
\[ \cos \theta = -\frac{1}{2} \]

7. **Finding the Angle**:
The angle \(\theta\) can be found using the inverse cosine function:
\[ \theta = \cos^{-1}(-\frac{1}{2}) \]
The angle whose cosine is \(-\frac{1}{2}\) is:
\[ \theta = 120^\circ \quad \text{(or } \theta = \frac{2\pi}{3} \text{ radians)} \]

### Conclusion:
The angle between the unit vectors \(\hat{e_1}\) and \(\hat{e_2}\) is \(120^\circ\).