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

Explanation

To solve the problem, we need to find the angle between two vectors \( \mathbf{A} \) and \( \mathbf{B} \) given that the magnitude of their sum is equal to the magnitude of their difference.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given that:
\[ |\mathbf{A} + \mathbf{B}| = |\mathbf{A} - \mathbf{B}| \]

2. **Using the Magnitude Formula**:
The magnitude of the sum of two vectors can be expressed as:
\[ |\mathbf{A} + \mathbf{B}| = \sqrt{|\mathbf{A}|^2 + |\mathbf{B}|^2 + 2|\mathbf{A}||\mathbf{B}|\cos\theta} \]
where \( \theta \) is the angle between the vectors \( \mathbf{A} \) and \( \mathbf{B} \).

Similarly, the magnitude of the difference of the two vectors is:
\[ |\mathbf{A} - \mathbf{B}| = \sqrt{|\mathbf{A}|^2 + |\mathbf{B}|^2 - 2|\mathbf{A}||\mathbf{B}|\cos\theta} \]

3. **Setting the Magnitudes Equal**:
Since we have:
\[ |\mathbf{A} + \mathbf{B}| = |\mathbf{A} - \mathbf{B}| \]
we can square both sides:
\[ |\mathbf{A}|^2 + |\mathbf{B}|^2 + 2|\mathbf{A}||\mathbf{B}|\cos\theta = |\mathbf{A}|^2 + |\mathbf{B}|^2 - 2|\mathbf{A}||\mathbf{B}|\cos\theta \]

4. **Simplifying the Equation**:
By cancelling \( |\mathbf{A}|^2 + |\mathbf{B}|^2 \) from both sides, we get:
\[ 2|\mathbf{A}||\mathbf{B}|\cos\theta = -2|\mathbf{A}||\mathbf{B}|\cos\theta \]

5. **Rearranging the Terms**:
This simplifies to:
\[ 2|\mathbf{A}||\mathbf{B}|\cos\theta + 2|\mathbf{A}||\mathbf{B}|\cos\theta = 0 \]
or:
\[ 4|\mathbf{A}||\mathbf{B}|\cos\theta = 0 \]

6. **Finding the Angle**:
Since \( |\mathbf{A}| \) and \( |\mathbf{B}| \) are magnitudes and cannot be zero (assuming non-zero vectors), we can conclude:
\[ \cos\theta = 0 \]
This implies:
\[ \theta = 90^\circ \]

### Final Answer:
The angle between the two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is \( 90^\circ \).