For any two vectors `veca and vecb` write when `| veca + vecb|= | veca-vecb|` holds.

Answer

To solve the problem where we need to determine when the magnitudes of the sum and difference of two vectors \( \vec{A} \) and \( \vec{B} \) are equal, we start with the equation:

\[ |\vec{A} + \vec{B}| = |\vec{A} - \vec{B}| \]

### Step 1: Square both sides
To eliminate the square root in the magnitudes, we square both sides of the equation:

\[ |\vec{A} + \vec{B}|^2 = |\vec{A} - \vec{B}|^2 \]

### Step 2: Expand both sides
Using the property of magnitudes, we expand both sides:

\[ (\vec{A} + \vec{B}) \cdot (\vec{A} + \vec{B}) = (\vec{A} - \vec{B}) \cdot (\vec{A} - \vec{B}) \]

This gives us:

\[ \vec{A} \cdot \vec{A} + 2 \vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{B} = \vec{A} \cdot \vec{A} - 2 \vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{B} \]

### Step 3: Simplify the equation
Now, we can simplify the equation by canceling out the common terms on both sides:

\[ 2 \vec{A} \cdot \vec{B} = -2 \vec{A} \cdot \vec{B} \]

### Step 4: Combine like terms
Combining the terms gives us:

\[ 2 \vec{A} \cdot \vec{B} + 2 \vec{A} \cdot \vec{B} = 0 \]

This simplifies to:

\[ 4 \vec{A} \cdot \vec{B} = 0 \]

### Step 5: Conclusion
From this equation, we can conclude that:

\[ \vec{A} \cdot \vec{B} = 0 \]

This implies that the vectors \( \vec{A} \) and \( \vec{B} \) are perpendicular to each other.

### Final Answer
The condition for \( |\vec{A} + \vec{B}| = |\vec{A} - \vec{B}| \) holds when the vectors \( \vec{A} \) and \( \vec{B} \) are perpendicular.

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