For two vectors `vecA and vecB, |vecA+vecB|= |vecA - vecB|` is always true when

Options

A. `|vec A| = |vecB |ne 0`
B. `vecAbotvecB`
C. `vec| A| =|vec B|ne 0 and vecA and vecB` are parallel
D. `vec| A| =|vec B|ne 0 and vecA and vecB` are antiparallel

Answer

To solve the problem where we need to determine when the condition \( |\vec{A} + \vec{B}| = |\vec{A} - \vec{B}| \) holds true, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Magnitude of Vectors**:
We start with the equation \( |\vec{A} + \vec{B}| = |\vec{A} - \vec{B}| \). This means that the magnitude of the vector sum of \( \vec{A} \) and \( \vec{B} \) is equal to the magnitude of the vector difference of \( \vec{A} \) and \( \vec{B} \).

2. **Using the Magnitude Formula**:
The magnitude of a vector can be expressed using the formula:
\[ |\vec{C}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}|\cos(\theta)} \]
where \( \theta \) is the angle between the vectors \( \vec{A} \) and \( \vec{B} \).

3. **Applying the Formula**:
For \( |\vec{A} + \vec{B}| \):
\[ |\vec{A} + \vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}|\cos(\theta) \]
For \( |\vec{A} - \vec{B}| \):
\[ |\vec{A} - \vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2 - 2|\vec{A}||\vec{B}|\cos(\theta) \]

4. **Setting the Equations Equal**:
Since we have \( |\vec{A} + \vec{B}|^2 = |\vec{A} - \vec{B}|^2 \), we can set the two equations equal:
\[ |\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}|\cos(\theta) = |\vec{A}|^2 + |\vec{B}|^2 - 2|\vec{A}||\vec{B}|\cos(\theta) \]

5. **Simplifying the Equation**:
Canceling \( |\vec{A}|^2 + |\vec{B}|^2 \) from both sides gives:
\[ 2|\vec{A}||\vec{B}|\cos(\theta) + 2|\vec{A}||\vec{B}|\cos(\theta) = 0 \]
This simplifies to:
\[ 4|\vec{A}||\vec{B}|\cos(\theta) = 0 \]

6. **Analyzing the Result**:
Since \( |\vec{A}| \) and \( |\vec{B}| \) are not zero (assuming both vectors are non-zero), we conclude that:
\[ \cos(\theta) = 0 \]
This implies that \( \theta = 90^\circ \). Therefore, the vectors \( \vec{A} \) and \( \vec{B} \) must be perpendicular to each other.

### Conclusion:
The condition \( |\vec{A} + \vec{B}| = |\vec{A} - \vec{B}| \) holds true when the vectors \( \vec{A} \) and \( \vec{B} \) are perpendicular to each other.