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

Options

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

Answer

To solve the problem, we need to analyze the relationship between the magnitudes of the sum and difference of two vectors, A and B, and the angle between them, θ.

### Step-by-Step Solution:

1. **Understand the Given Condition**: We are given that the magnitude of the sum of two vectors is equal to the magnitude of their difference. Mathematically, this can be expressed as:
\[ |A + B| = |A - B| \]

2. **Use the Formula for Magnitudes**: We can express the magnitudes of the sum and difference of the vectors using the formula:
\[ |A + B| = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]
\[ |A - B| = \sqrt{A^2 + B^2 - 2AB \cos \theta} \]

3. **Set the Magnitudes Equal**: Since we know that the magnitudes are equal, we can set the two equations equal to each other:
\[ \sqrt{A^2 + B^2 + 2AB \cos \theta} = \sqrt{A^2 + B^2 - 2AB \cos \theta} \]

4. **Square Both Sides**: To eliminate the square roots, we square both sides:
\[ A^2 + B^2 + 2AB \cos \theta = A^2 + B^2 - 2AB \cos \theta \]

5. **Simplify the Equation**: We can simplify the equation by subtracting \(A^2 + B^2\) from both sides:
\[ 2AB \cos \theta = -2AB \cos \theta \]

6. **Combine Like Terms**: Adding \(2AB \cos \theta\) to both sides gives:
\[ 2AB \cos \theta + 2AB \cos \theta = 0 \]
\[ 4AB \cos \theta = 0 \]

7. **Solve for cos θ**: Since \(A\) and \(B\) are non-zero vectors, we can divide both sides by \(4AB\):
\[ \cos \theta = 0 \]

8. **Find the Angle θ**: The cosine of an angle is zero at:
\[ \theta = 90^\circ \]

### Final Answer:
The angle between the two vectors is \(90^\circ\).