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 \( \vec{A} \) and \( \vec{B} \) given that the resultant vector \( \vec{R} \) is defined by the equation \( |\vec{R}| = |\vec{A}| - |\vec{B}| \), we can follow these steps:

### Step 1: Understand the Resultant Vector
The resultant vector \( \vec{R} \) can be expressed using the formula:
\[ |\vec{R}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos \theta} \]
where \( \theta \) is the angle between vectors \( \vec{A} \) and \( \vec{B} \).

### Step 2: Set Up the Equation
According to the problem, we know:
\[ |\vec{R}| = |\vec{A}| - |\vec{B}| \]
We can square both sides to eliminate the square root:
\[ (|\vec{A}| - |\vec{B}|)^2 = |\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos \theta \]

### Step 3: Expand the Left Side
Expanding the left side gives:
\[ |\vec{A}|^2 - 2|\vec{A}||\vec{B}| + |\vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos \theta \]

### Step 4: Simplify the Equation
Now, we can cancel \( |\vec{A}|^2 \) and \( |\vec{B}|^2 \) from both sides:
\[ -2|\vec{A}||\vec{B}| = 2 |\vec{A}| |\vec{B}| \cos \theta \]

### Step 5: Solve for \( \cos \theta \)
Dividing both sides by \( 2 |\vec{A}||\vec{B}| \) (assuming \( |\vec{A}| \) and \( |\vec{B}| \) are not zero):
\[ -1 = \cos \theta \]

### Step 6: Find the Angle
The angle \( \theta \) that satisfies \( \cos \theta = -1 \) is:
\[ \theta = 180^\circ \]

### Conclusion
Thus, the angle between vectors \( \vec{A} \) and \( \vec{B} \) is \( 180^\circ \).

### Final Answer
The angle between \( \vec{A} \) and \( \vec{B} \) is \( 180^\circ \).

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