Two vectors `vecA` and `vecB` are such that `vecA+vecB=vecC` and `A^(2)+B^(2)=C^(2)`. Which of the following statements, is correct:-

Options

A. A is parallel to B
B. A is antiparallel to B
C. A is perpendicular to B
D. A and B are equal in magnitude

Explanation

To solve the problem, we need to analyze the given equations involving the vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\):

1. **Given Equations**:
- \(\vec{A} + \vec{B} = \vec{C}\)
- \(A^2 + B^2 = C^2\)

2. **Understanding the Equations**:
- The first equation states that the vector sum of \(\vec{A}\) and \(\vec{B}\) results in \(\vec{C}\).
- The second equation resembles the Pythagorean theorem, which suggests a relationship between the magnitudes of the vectors.

3. **Using the Dot Product**:
- We can take the dot product of both sides of the first equation:
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} + \vec{B}) = \vec{C} \cdot \vec{C} \]
- Expanding the left-hand side using the distributive property of the dot product:
\[ \vec{A} \cdot \vec{A} + 2 \vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{B} = \vec{C} \cdot \vec{C} \]
- This simplifies to:
\[ A^2 + 2 \vec{A} \cdot \vec{B} + B^2 = C^2 \]

4. **Substituting the Second Equation**:
- From the second equation, we know that \(A^2 + B^2 = C^2\). We can substitute this into our expanded equation:
\[ C^2 + 2 \vec{A} \cdot \vec{B} = C^2 \]
- By subtracting \(C^2\) from both sides, we get:
\[ 2 \vec{A} \cdot \vec{B} = 0 \]

5. **Conclusion about the Dot Product**:
- Since \(2 \vec{A} \cdot \vec{B} = 0\), it follows that:
\[ \vec{A} \cdot \vec{B} = 0 \]
- This means that the vectors \(\vec{A}\) and \(\vec{B}\) are perpendicular to each other.

6. **Final Answer**:
- Therefore, the correct statement is that \(\vec{A}\) is perpendicular to \(\vec{B}\).