If A and B are Two non -zero vector having equal magnitude , the angle between the vector A and A-B is

Options

A. `0^(@)`
B. `90^(@)`
C. `180^(@)`
D. dependent on the orientation o fA and B

Explanation

To solve the problem of finding the angle between the vector \( \mathbf{A} \) and \( \mathbf{A} - \mathbf{B} \), given that \( \mathbf{A} \) and \( \mathbf{B} \) are two non-zero vectors with equal magnitudes, we can follow these steps:

### Step 1: Understand the Given Information
We know that:
- \( |\mathbf{A}| = |\mathbf{B}| \) (the magnitudes of vectors A and B are equal).
- We need to find the angle \( \phi \) between \( \mathbf{A} \) and \( \mathbf{A} - \mathbf{B} \).

### Step 2: Use the Cosine Rule
To find the angle between \( \mathbf{A} \) and \( \mathbf{A} - \mathbf{B} \), we can use the cosine rule in the triangle formed by the vectors \( \mathbf{A} \), \( \mathbf{B} \), and \( \mathbf{A} - \mathbf{B} \).

The cosine rule states:
\[ c^2 = a^2 + b^2 - 2ab \cos(\theta) \]
In our case:
- Let \( c = |\mathbf{A} - \mathbf{B}| \)
- Let \( a = |\mathbf{A}| \)
- Let \( b = |\mathbf{B}| \)

Since \( |\mathbf{A}| = |\mathbf{B}| \), we can denote this common magnitude as \( A \). Thus, we have:
\[ |\mathbf{A} - \mathbf{B}|^2 = A^2 + A^2 - 2A^2 \cos(\theta) = 2A^2(1 - \cos(\theta)) \]

### Step 3: Calculate the Magnitude of \( \mathbf{A} - \mathbf{B} \)
The magnitude of \( \mathbf{A} - \mathbf{B} \) can also be expressed as:
\[ |\mathbf{A} - \mathbf{B}| = \sqrt{|\mathbf{A}|^2 + |\mathbf{B}|^2 - 2|\mathbf{A}||\mathbf{B}|\cos(\theta)} = \sqrt{A^2 + A^2 - 2A^2 \cos(\theta)} = \sqrt{2A^2(1 - \cos(\theta))} \]

### Step 4: Find the Angle Between \( \mathbf{A} \) and \( \mathbf{A} - \mathbf{B} \)
Let \( \phi \) be the angle between \( \mathbf{A} \) and \( \mathbf{A} - \mathbf{B} \). Using the dot product:
\[ \mathbf{A} \cdot (\mathbf{A} - \mathbf{B}) = |\mathbf{A}||\mathbf{A} - \mathbf{B}|\cos(\phi) \]
Substituting the values we have:
\[ |\mathbf{A}|^2 - \mathbf{A} \cdot \mathbf{B} = A |\mathbf{A} - \mathbf{B}|\cos(\phi) \]

### Step 5: Simplify and Solve for \( \phi \)
Using the fact that \( \mathbf{A} \cdot \mathbf{B} = A^2 \cos(\theta) \):
\[ A^2 - A^2 \cos(\theta) = A |\mathbf{A} - \mathbf{B}|\cos(\phi) \]

From earlier, we know:
\[ |\mathbf{A} - \mathbf{B}| = \sqrt{2A^2(1 - \cos(\theta))} \]
Substituting this back, we can find \( \phi \).

### Conclusion
After simplification, we find that:
\[ \phi = 90^\circ - \frac{\theta}{2} \]
This shows that the angle between \( \mathbf{A} \) and \( \mathbf{A} - \mathbf{B} \) depends on the angle \( \theta \) between \( \mathbf{A} \) and \( \mathbf{B} \).