Statement 1: If `| vec a+ vec b|=| vec a- vec b|,t h e n vec aa n d vec b` are perpendicular to each other. Statement 2: If the diagonal of a parallelogram are equal magnitude, then the parallelogram is a rectangle.

Options

A. Statement-II and statement II ar correct and Statement III is the correct explanation of statement I
B. Both statement I and statement II are correct but statement II is not the correct explanation of statement I
C. Statement I is correct but statement II is incorrect
D. Statement II is correct but statement I is incorrect

Explanation

To solve the given question, we need to analyze both statements one by one.

### Statement 1:
If \( |\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \), then \(\vec{a}\) and \(\vec{b}\) are perpendicular to each other.

**Step 1: Start with the given equation.**
\[ |\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \]

**Step 2: Square both sides to eliminate the modulus.**
\[ |\vec{a} + \vec{b}|^2 = |\vec{a} - \vec{b}|^2 \]

**Step 3: Expand both sides using the formula \( | \vec{x} |^2 = \vec{x} \cdot \vec{x} \).**
\[ (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b}) \]

**Step 4: Simplify both sides.**
\[ \vec{a} \cdot \vec{a} + 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = \vec{a} \cdot \vec{a} - 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} \]

**Step 5: Cancel out common terms.**
\[ 2 \vec{a} \cdot \vec{b} = -2 \vec{a} \cdot \vec{b} \]

**Step 6: Combine like terms.**
\[ 4 \vec{a} \cdot \vec{b} = 0 \]

**Step 7: Conclude that \( \vec{a} \cdot \vec{b} = 0 \).**
This implies that \(\vec{a}\) and \(\vec{b}\) are perpendicular.

### Conclusion for Statement 1:
Statement 1 is **true**.

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### Statement 2:
If the diagonals of a parallelogram are equal in magnitude, then the parallelogram is a rectangle.

**Step 1: Define the parallelogram and its diagonals.**
Let \(ABCD\) be a parallelogram with diagonals \(AC\) and \(BD\).

**Step 2: Express the diagonals using vectors.**
The diagonal \(AC\) can be expressed as:
\[ \vec{AC} = \vec{A} + \vec{B} \]
The diagonal \(BD\) can be expressed as:
\[ \vec{BD} = \vec{B} - \vec{A} \]

**Step 3: Set the magnitudes of the diagonals equal.**
Given that the diagonals are equal in magnitude:
\[ |\vec{AC}| = |\vec{BD}| \]

**Step 4: Square both sides to eliminate the modulus.**
\[ |\vec{A} + \vec{B}|^2 = |\vec{B} - \vec{A}|^2 \]

**Step 5: Expand both sides.**
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} + \vec{B}) = (\vec{B} - \vec{A}) \cdot (\vec{B} - \vec{A}) \]

**Step 6: Simplify both sides.**
\[ \vec{A} \cdot \vec{A} + 2 \vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{B} = \vec{B} \cdot \vec{B} - 2 \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{A} \]

**Step 7: Cancel out common terms.**
\[ 2 \vec{A} \cdot \vec{B} = -2 \vec{A} \cdot \vec{B} \]

**Step 8: Combine like terms.**
\[ 4 \vec{A} \cdot \vec{B} = 0 \]

**Step 9: Conclude that \( \vec{A} \cdot \vec{B} = 0 \).**
This implies that the angles between the sides of the parallelogram are \(90^\circ\), indicating that \(ABCD\) is a rectangle.

### Conclusion for Statement 2:
Statement 2 is **true**.

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### Final Conclusion:
Both statements are true.