Two vectors `vec A and vecB` are joined with their tails at the same position . A parallelogram is completed using these two vectors . It is found that both the diagonals of this parallelogram are perpendicular to each other . Select the correct option .

Options

A. `A=4B`
B. `A=3B`
C. `A=2B`
D. `A=B`

Explanation

To solve the problem, we need to analyze the situation where two vectors \(\vec{A}\) and \(\vec{B}\) are joined at the same point to form a parallelogram, and we know that the diagonals of this parallelogram are perpendicular to each other.

### Step-by-step Solution:

1. **Understanding the Parallelogram**:
When two vectors \(\vec{A}\) and \(\vec{B}\) are placed tail to tail, they form a parallelogram. The diagonals of the parallelogram can be represented as:
- Diagonal 1: \(\vec{D_1} = \vec{A} + \vec{B}\)
- Diagonal 2: \(\vec{D_2} = \vec{A} - \vec{B}\)

2. **Condition of Perpendicular Diagonals**:
The problem states that these diagonals are perpendicular to each other. For two vectors to be perpendicular, their dot product must be zero:
\[ \vec{D_1} \cdot \vec{D_2} = 0 \]

3. **Calculating the Dot Product**:
Substitute the expressions for the diagonals into the dot product:
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B}) = 0 \]

4. **Expanding the Dot Product**:
Using the distributive property of the dot product:
\[ \vec{A} \cdot \vec{A} - \vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{A} - \vec{B} \cdot \vec{B} = 0 \]
Since \(\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}\), we can simplify this to:
\[ |\vec{A}|^2 - |\vec{B}|^2 = 0 \]

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

### Final Answer:
The correct option is that \( |\vec{A}| = |\vec{B}| \).