The vector sum of two forces is perpendicular to their vector differences. In that case, the forces

Options

A. are not equal to each other in magnitude
B. are parallel
C. are perpendicular
D. are rqual to each other in magnitude

Explanation

To solve the problem, we need to analyze the given condition that the vector sum of two forces is perpendicular to their vector difference. Let's denote the two forces as vectors **A** and **B**.

### Step-by-Step Solution:

1. **Understanding the Condition**:
We are given that the vector sum of two forces is perpendicular to their vector difference. Mathematically, this can be expressed as:
\[ (\mathbf{A} + \mathbf{B}) \perp (\mathbf{A} - \mathbf{B}) \]

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

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

4. **Setting the Equation**:
This simplifies to:
\[ |\mathbf{A}|^2 = |\mathbf{B}|^2 \]
Taking the square root of both sides gives us:
\[ |\mathbf{A}| = |\mathbf{B}| \]

5. **Conclusion**:
The magnitudes of the two forces (vectors) are equal. Therefore, the correct answer is that the forces are equal in magnitude.

### Final Answer:
The forces are equal to each other in magnitude.

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