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. A and B have the same direction

Explanation

To solve the problem, we need to analyze the given condition that 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:
\[ \mathbf{A} + \mathbf{B} \perp \mathbf{A} - \mathbf{B} \]
This means that the dot product of these two vectors is zero:
\[ (\mathbf{A} + \mathbf{B}) \cdot (\mathbf{A} - \mathbf{B}) = 0 \]

2. **Expanding the Dot Product**:
We can expand the left-hand side using the distributive property of the dot product:
\[ \mathbf{A} \cdot \mathbf{A} - \mathbf{A} \cdot \mathbf{B} + \mathbf{B} \cdot \mathbf{A} - \mathbf{B} \cdot \mathbf{B} = 0 \]
Since \( \mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A} \), we can simplify this to:
\[ |\mathbf{A}|^2 - |\mathbf{B}|^2 = 0 \]

3. **Setting Up the Equation**:
From the equation \( |\mathbf{A}|^2 - |\mathbf{B}|^2 = 0 \), we can rearrange it to:
\[ |\mathbf{A}|^2 = |\mathbf{B}|^2 \]
This implies that the magnitudes of the vectors are equal:
\[ |\mathbf{A}| = |\mathbf{B}| \]

4. **Conclusion**:
The equality of magnitudes means that the two vectors can be equal, i.e., \( \mathbf{A} = \mathbf{B} \) or they can be equal in magnitude but opposite in direction, i.e., \( \mathbf{A} = -\mathbf{B} \). However, since we are looking for the options provided in the question, the only option that is definitively true is:
\[ \mathbf{A} = \mathbf{B} \]

### Final Answer:
The correct option is:
- Option 1: \( \mathbf{A} = \mathbf{B} \)