If the magnitude of sum of two vectors is equal to the magnitude of difference of the two vector, the angle between these Vector is

Options

A. `90^(@)`
B. `45^(@)`
C. `180^(@)`
D. `0^(@)`

Explanation

To solve the problem, we need to establish the relationship between the magnitudes of the sum and difference of two vectors and find the angle between them. Let's denote the two vectors as **A** and **B**.

### Step-by-Step Solution:

1. **Write the equation for the magnitudes**:
According to the problem, we have:
\[ |\mathbf{A} + \mathbf{B}| = |\mathbf{A} - \mathbf{B}| \]

2. **Square both sides to eliminate the square root**:
Squaring both sides gives us:
\[ |\mathbf{A} + \mathbf{B}|^2 = |\mathbf{A} - \mathbf{B}|^2 \]

3. **Expand both sides using the formula for the magnitude of a vector**:
The magnitude squared of a vector can be expressed as:
\[ |\mathbf{A} + \mathbf{B}|^2 = \mathbf{A} \cdot \mathbf{A} + \mathbf{B} \cdot \mathbf{B} + 2 \mathbf{A} \cdot \mathbf{B} \]
\[ |\mathbf{A} - \mathbf{B}|^2 = \mathbf{A} \cdot \mathbf{A} + \mathbf{B} \cdot \mathbf{B} - 2 \mathbf{A} \cdot \mathbf{B} \]

4. **Set the expanded forms equal to each other**:
\[ \mathbf{A} \cdot \mathbf{A} + \mathbf{B} \cdot \mathbf{B} + 2 \mathbf{A} \cdot \mathbf{B} = \mathbf{A} \cdot \mathbf{A} + \mathbf{B} \cdot \mathbf{B} - 2 \mathbf{A} \cdot \mathbf{B} \]

5. **Cancel out common terms**:
The terms \(\mathbf{A} \cdot \mathbf{A}\) and \(\mathbf{B} \cdot \mathbf{B}\) cancel out:
\[ 2 \mathbf{A} \cdot \mathbf{B} = -2 \mathbf{A} \cdot \mathbf{B} \]

6. **Combine like terms**:
\[ 2 \mathbf{A} \cdot \mathbf{B} + 2 \mathbf{A} \cdot \mathbf{B} = 0 \]
\[ 4 \mathbf{A} \cdot \mathbf{B} = 0 \]

7. **Solve for the dot product**:
Since \(4 \mathbf{A} \cdot \mathbf{B} = 0\), we have:
\[ \mathbf{A} \cdot \mathbf{B} = 0 \]

8. **Interpret the result**:
The dot product of two vectors is zero when the angle \(\theta\) between them is \(90^\circ\) (or \(\frac{\pi}{2}\) radians).

### Conclusion:
Thus, the angle between the two vectors **A** and **B** is:
\[ \theta = 90^\circ \]