Statement 1 : If `|veca + vecb| = |veca - vecb|`, then `veca and vecb` are perpendicular to each other.
Statement 2 : If the diagonals of a parallelogram are equal in magnitude, then the parallelogram is a rectangle.

Options

A. Both the statements are true, and Statement 2 is the correct explanation for Statement 1.
B. Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.
C. Statement 1 is true and Statement 2 is false.
D. Statement 1 is false and Statement 2 is true.

Explanation

To solve the problem, we will analyze both statements step by step.

### Step 1: Analyzing Statement 1
We start with the statement:
**If \( |\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \), then \(\vec{a}\) and \(\vec{b}\) are perpendicular to each other.**

1. **Square both sides**:
\[ |\vec{a} + \vec{b}|^2 = |\vec{a} - \vec{b}|^2 \]
This leads to:
\[ (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b}) \]

2. **Expand both sides**:
\[ \vec{a} \cdot \vec{a} + 2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = \vec{a} \cdot \vec{a} - 2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} \]

3. **Simplify the equation**:
\[ 2\vec{a} \cdot \vec{b} = -2\vec{a} \cdot \vec{b} \]

4. **Combine like terms**:
\[ 4\vec{a} \cdot \vec{b} = 0 \]

5. **Conclusion**:
Since \(4\vec{a} \cdot \vec{b} = 0\), it follows that \(\vec{a} \cdot \vec{b} = 0\), which means \(\vec{a}\) and \(\vec{b}\) are perpendicular.

### Step 2: Analyzing Statement 2
Now we analyze the second statement:
**If the diagonals of a parallelogram are equal in magnitude, then the parallelogram is a rectangle.**

1. **Recall the properties of a parallelogram**:
In a parallelogram, the diagonals bisect each other.

2. **Let the diagonals be represented as vectors**:
Let \(\vec{d_1}\) and \(\vec{d_2}\) be the diagonals of the parallelogram.

3. **If the diagonals are equal**:
\[ |\vec{d_1}| = |\vec{d_2}| \]
This implies that the sides of the parallelogram must be equal in length and angles between the sides must be \(90^\circ\) for the diagonals to be equal.

4. **Conclusion**:
Therefore, if the diagonals of a parallelogram are equal, it must be a rectangle.

### Final Conclusion
Both statements are true. Statement 2 is the correct explanation for Statement 1.

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