If A and B are two vectors such that |A+B|=2|A-B|. The angle between vectors A and B is

Options

A. `45^(@)`
B. `60^(@)`
C. `30^(@)`
D. data insuffcient

Explanation

To solve the problem, we need to find the angle between two vectors \( \vec{A} \) and \( \vec{B} \) given the condition that \( |\vec{A} + \vec{B}| = 2 |\vec{A} - \vec{B}| \).

### Step-by-Step Solution:

1. **Write down the given equation**:
\[ |\vec{A} + \vec{B}| = 2 |\vec{A} - \vec{B}| \]

2. **Square both sides** to eliminate the square root:
\[ |\vec{A} + \vec{B}|^2 = 4 |\vec{A} - \vec{B}|^2 \]

3. **Use the formula for the magnitude of a vector**:
\[ |\vec{A} + \vec{B}|^2 = \vec{A} \cdot \vec{A} + \vec{B} \cdot \vec{B} + 2 \vec{A} \cdot \vec{B} \]
\[ |\vec{A} - \vec{B}|^2 = \vec{A} \cdot \vec{A} + \vec{B} \cdot \vec{B} - 2 \vec{A} \cdot \vec{B} \]

4. **Substituting these into the equation**:
\[ \vec{A} \cdot \vec{A} + \vec{B} \cdot \vec{B} + 2 \vec{A} \cdot \vec{B} = 4 \left( \vec{A} \cdot \vec{A} + \vec{B} \cdot \vec{B} - 2 \vec{A} \cdot \vec{B} \right) \]

5. **Expand the right-hand side**:
\[ \vec{A}^2 + \vec{B}^2 + 2 \vec{A} \cdot \vec{B} = 4 \vec{A}^2 + 4 \vec{B}^2 - 8 \vec{A} \cdot \vec{B} \]

6. **Rearranging the equation**:
\[ 0 = 4 \vec{A}^2 + 4 \vec{B}^2 - 8 \vec{A} \cdot \vec{B} - \vec{A}^2 - \vec{B}^2 - 2 \vec{A} \cdot \vec{B} \]
\[ 0 = 3 \vec{A}^2 + 3 \vec{B}^2 - 10 \vec{A} \cdot \vec{B} \]

7. **Dividing the whole equation by 3**:
\[ 0 = \vec{A}^2 + \vec{B}^2 - \frac{10}{3} \vec{A} \cdot \vec{B} \]

8. **Using the dot product in terms of the angle**:
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \]
Substitute this into the equation:
\[ 0 = |\vec{A}|^2 + |\vec{B}|^2 - \frac{10}{3} |\vec{A}| |\vec{B}| \cos \theta \]

9. **Rearranging for \( \cos \theta \)**:
\[ \frac{10}{3} |\vec{A}| |\vec{B}| \cos \theta = |\vec{A}|^2 + |\vec{B}|^2 \]
\[ \cos \theta = \frac{3 (|\vec{A}|^2 + |\vec{B}|^2)}{10 |\vec{A}| |\vec{B}|} \]

10. **Finding the angle \( \theta \)**:
Since the specific magnitudes of \( |\vec{A}| \) and \( |\vec{B}| \) are not provided, we cannot determine a specific numerical value for \( \theta \). Therefore, we conclude that the data is insufficient to find a unique angle.

### Conclusion:
The angle between vectors \( \vec{A} \) and \( \vec{B} \) cannot be determined uniquely with the given information, hence the correct option is **data insufficient**.