If `|overset(rarr)A-overset(rarr)B|=|overset(rarr)A|-|overset(rarr)B|` the angle between `overset(rarr)A` and `overset(rarr)B` is

Options

A. `60^(@)`
B. `0^(@)`
C. `120^(@)`
D. `90^(@)`

Explanation

To solve the problem, we need to analyze the equation given:

\[ |\vec{A} - \vec{B}| = |\vec{A}| - |\vec{B}| \]

### Step 1: Square both sides of the equation

We start by squaring both sides of the equation to eliminate the square root:

\[ |\vec{A} - \vec{B}|^2 = (|\vec{A}| - |\vec{B}|)^2 \]

### Step 2: Expand both sides

Using the properties of magnitudes and the formula for the magnitude of the difference of two vectors, we can expand both sides:

Left side:
\[ |\vec{A} - \vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2 - 2|\vec{A}||\vec{B}|\cos\theta \]

Right side:
\[ (|\vec{A}| - |\vec{B}|)^2 = |\vec{A}|^2 - 2|\vec{A}||\vec{B}| + |\vec{B}|^2 \]

### Step 3: Set the two expansions equal to each other

Now, we set the left side equal to the right side:

\[ |\vec{A}|^2 + |\vec{B}|^2 - 2|\vec{A}||\vec{B}|\cos\theta = |\vec{A}|^2 - 2|\vec{A}||\vec{B}| + |\vec{B}|^2 \]

### Step 4: Simplify the equation

We can simplify this equation by canceling out \( |\vec{A}|^2 \) and \( |\vec{B}|^2 \) from both sides:

\[ -2|\vec{A}||\vec{B}|\cos\theta = -2|\vec{A}||\vec{B}| \]

### Step 5: Divide both sides by \(-2|\vec{A}||\vec{B}|\)

Assuming \( |\vec{A}| \) and \( |\vec{B}| \) are not zero, we can divide both sides by \(-2|\vec{A}||\vec{B}|\):

\[ \cos\theta = 1 \]

### Step 6: Find the angle \(\theta\)

The cosine of an angle is equal to 1 when the angle is:

\[ \theta = 0^\circ \]

### Conclusion

Thus, the angle between the vectors \(\vec{A}\) and \(\vec{B}\) is:

\[ \theta = 0^\circ \]

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