Resultant of two vetors A and B is given by `|R|={|A|-|B|}.`angle between A and B will be

Options

A. `90^(@)`
B. `180^(@)`
C. `0^(@)`
D. none of these

Explanation

To find the angle between two vectors A and B given that the magnitude of their resultant R is given by the equation |R| = |A| - |B|, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
We know that the resultant of two vectors A and B is given by the equation:
\[ |R| = |A| - |B| \]

2. **Use the Formula for the Magnitude of the Resultant**:
The magnitude of the resultant of two vectors A and B, which makes an angle θ with each other, is given by:
\[ |R| = \sqrt{|A|^2 + |B|^2 + 2|A||B|\cos\theta} \]

3. **Set the Two Expressions for |R| Equal**:
Since both expressions represent the magnitude of the resultant, we can equate them:
\[ \sqrt{|A|^2 + |B|^2 + 2|A||B|\cos\theta} = |A| - |B| \]

4. **Square Both Sides to Eliminate the Square Root**:
Squaring both sides gives:
\[ |A|^2 + |B|^2 + 2|A||B|\cos\theta = (|A| - |B|)^2 \]

5. **Expand the Right Side**:
Expanding the right side:
\[ (|A| - |B|)^2 = |A|^2 - 2|A||B| + |B|^2 \]

6. **Set the Equations Equal**:
Now we have:
\[ |A|^2 + |B|^2 + 2|A||B|\cos\theta = |A|^2 - 2|A||B| + |B|^2 \]

7. **Simplify the Equation**:
Cancel out |A|^2 and |B|^2 from both sides:
\[ 2|A||B|\cos\theta = -2|A||B| \]

8. **Divide by 2|A||B| (assuming |A| and |B| are not zero)**:
\[ \cos\theta = -1 \]

9. **Find the Angle θ**:
The angle θ for which cosθ = -1 is:
\[ \theta = 180^\circ \]

### Final Answer:
The angle between vectors A and B is \( 180^\circ \).

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