If for two vector `vecA` and `vecB`, sum `(vecA+vecB)` is perpendicular to the difference `(vecA-vecB)`. The ratio of their magnitude is

Options

A. 1
B. 2
C. 3
D. none of these

Explanation

To solve the problem, we need to determine the ratio of the magnitudes of two vectors \(\vec{A}\) and \(\vec{B}\) given that their sum is perpendicular to their difference.

### Step-by-Step Solution:

1. **Understanding the Condition of Perpendicularity**:
We know that two vectors \(\vec{A} + \vec{B}\) and \(\vec{A} - \vec{B}\) are perpendicular. This means that their dot product is zero:
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B}) = 0 \]

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

3. **Using the Commutative Property**:
Since the dot product is commutative (\(\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}\)), we can simplify the equation:
\[ \vec{A} \cdot \vec{A} - \vec{B} \cdot \vec{B} = 0 \]
This simplifies to:
\[ |\vec{A}|^2 - |\vec{B}|^2 = 0 \]

4. **Rearranging the Equation**:
Rearranging the equation gives us:
\[ |\vec{A}|^2 = |\vec{B}|^2 \]

5. **Taking the Square Root**:
Taking the square root of both sides, we find:
\[ |\vec{A}| = |\vec{B}| \]

6. **Finding the Ratio**:
The ratio of the magnitudes of \(\vec{A}\) and \(\vec{B}\) is:
\[ \frac{|\vec{A}|}{|\vec{B}|} = 1 \]

### Conclusion:
Thus, the ratio of the magnitudes of the vectors \(\vec{A}\) and \(\vec{B}\) is \(1\).