Two vector `vec(A)` and `vec(B)` have equal magnitudes. Then the vector `vec(A) + vec(B)` is perpendicular :

Options

A. `vec(A) xx vec(B)`
B. `vec(A) - vec(B)`
C. `3vec(A) - 3vec(B)`
D. All of these

Answer

To solve the problem, we need to determine if the vector sum of two equal magnitude vectors \( \vec{A} \) and \( \vec{B} \) is perpendicular to another vector. We will analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Vectors**:
Given that \( |\vec{A}| = |\vec{B}| \), we can denote their common magnitude as \( k \). Thus, we have:
\[ |\vec{A}| = k \quad \text{and} \quad |\vec{B}| = k \]

2. **Finding the Resultant Vector**:
The resultant vector \( \vec{R} \) is given by:
\[ \vec{R} = \vec{A} + \vec{B} \]

3. **Using the Dot Product for Perpendicularity**:
To check if \( \vec{R} \) is perpendicular to \( \vec{A} - \vec{B} \), we can use the dot product:
\[ \vec{R} \cdot (\vec{A} - \vec{B}) = 0 \]

4. **Calculating the Dot Product**:
Substitute \( \vec{R} \) into the dot product:
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B}) = \vec{A} \cdot \vec{A} - \vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{A} - \vec{B} \cdot \vec{B} \]
Simplifying this gives:
\[ |\vec{A}|^2 - |\vec{B}|^2 + \vec{A} \cdot \vec{B} - \vec{B} \cdot \vec{A} \]

5. **Substituting Magnitudes**:
Since \( |\vec{A}| = |\vec{B}| = k \):
\[ |\vec{A}|^2 = k^2 \quad \text{and} \quad |\vec{B}|^2 = k^2 \]
Therefore, we have:
\[ k^2 - k^2 + \vec{A} \cdot \vec{B} - \vec{B} \cdot \vec{A} = 0 \]
This simplifies to:
\[ 0 + 0 = 0 \]

6. **Conclusion**:
Since the dot product \( \vec{R} \cdot (\vec{A} - \vec{B}) = 0 \), it follows that \( \vec{R} \) is indeed perpendicular to \( \vec{A} - \vec{B} \).

### Final Answer:
The vector \( \vec{A} + \vec{B} \) is perpendicular to \( \vec{A} - \vec{B} \).