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

Options

A. a. Equal to each other
B. b. Equal to each other in a magnitude
C. c. Not equal to each other in magnitude
D. d. Cannot be predicted

Explanation

To solve the problem, we need to analyze the given information about the vectors A and B.

### Step-by-Step Solution:

1. **Define the Vectors**:
Let vector \( \mathbf{A} \) and vector \( \mathbf{B} \) be the two vectors in question.

2. **Sum and Difference of Vectors**:
- The sum of the vectors is given by:
\[ \mathbf{P} = \mathbf{A} + \mathbf{B} \]
- The difference of the vectors is given by:
\[ \mathbf{Q} = \mathbf{A} - \mathbf{B} \]

3. **Condition of Perpendicularity**:
According to the problem, the sum \( \mathbf{P} \) is perpendicular to the difference \( \mathbf{Q} \). This means:
\[ \mathbf{P} \cdot \mathbf{Q} = 0 \]

4. **Substituting the Expressions for P and Q**:
Substitute \( \mathbf{P} \) and \( \mathbf{Q} \) into the dot product:
\[ (\mathbf{A} + \mathbf{B}) \cdot (\mathbf{A} - \mathbf{B}) = 0 \]

5. **Expanding the Dot Product**:
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:
\[ |\mathbf{A}|^2 - |\mathbf{B}|^2 = 0 \]

6. **Conclusion**:
From the equation \( |\mathbf{A}|^2 - |\mathbf{B}|^2 = 0 \), we can conclude that:
\[ |\mathbf{A}|^2 = |\mathbf{B}|^2 \implies |\mathbf{A}| = |\mathbf{B}| \]
This means that the magnitudes of vectors \( \mathbf{A} \) and \( \mathbf{B} \) are equal.

### Final Answer:
The correct option is that the magnitudes of vectors \( \mathbf{A} \) and \( \mathbf{B} \) are equal.

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