The sum of two vectors A and B is at right angles to their difference. Then

Options

A. A=B
B. A=2B
C. B=2A
D. None of these

Explanation

To solve the problem, we need to analyze the condition given in the question: the sum of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is at right angles to their difference.

### Step-by-Step Solution:

1. **Understanding the Condition**:
We are given that the sum of the vectors \( \mathbf{A} + \mathbf{B} \) is perpendicular to their difference \( \mathbf{A} - \mathbf{B} \).

2. **Using the Dot Product**:
For two vectors to be perpendicular, their dot product must be zero. Therefore, we can write:
\[ (\mathbf{A} + \mathbf{B}) \cdot (\mathbf{A} - \mathbf{B}) = 0 \]

3. **Expanding the Dot Product**:
We can expand the left-hand side using the distributive property of the dot product:
\[ \mathbf{A} \cdot \mathbf{A} - \mathbf{A} \cdot \mathbf{B} + \mathbf{B} \cdot \mathbf{A} - \mathbf{B} \cdot \mathbf{B} = 0 \]

4. **Simplifying the Expression**:
We know that \( \mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A} \). Thus, we can combine the terms:
\[ \mathbf{A} \cdot \mathbf{A} - \mathbf{B} \cdot \mathbf{B} = 0 \]
This implies:
\[ \mathbf{A} \cdot \mathbf{A} = \mathbf{B} \cdot \mathbf{B} \]

5. **Relating Magnitudes**:
The dot product \( \mathbf{A} \cdot \mathbf{A} \) is equal to the square of the magnitude of vector \( \mathbf{A} \), and similarly for vector \( \mathbf{B} \):
\[ |\mathbf{A}|^2 = |\mathbf{B}|^2 \]

6. **Conclusion**:
From the above equation, we can conclude that:
\[ |\mathbf{A}| = |\mathbf{B}| \]
This means that the magnitudes of vectors \( \mathbf{A} \) and \( \mathbf{B} \) are equal, which implies that:
\[ \mathbf{A} = \mathbf{B} \]
Therefore, the correct option is:
\[ \text{Option 1: } \mathbf{A} = \mathbf{B} \]