If vector A and B have an angle `theta` between them, then value of `|hatA -hatB|` will be

Options

A. `2sin""(theta)/(2)`
B. `2cos""(theta)/(2)`
C. `2tan""(theta)/(2)`
D. `tantheta`

Explanation

To solve the problem of finding the magnitude of the difference between two unit vectors \(\hat{A}\) and \(\hat{B}\) that have an angle \(\theta\) between them, we can follow these steps:

### Step 1: Understand the Magnitude of the Difference
We need to find the magnitude of the vector difference \(|\hat{A} - \hat{B}|\).

### Step 2: Use the Formula for the Magnitude of a Vector Difference
The magnitude of the difference of two vectors can be calculated using the formula:
\[ |\hat{A} - \hat{B}| = \sqrt{|\hat{A}|^2 + |\hat{B}|^2 - 2|\hat{A}||\hat{B}|\cos(\theta)} \]

### Step 3: Substitute the Magnitudes of Unit Vectors
Since \(\hat{A}\) and \(\hat{B}\) are unit vectors, their magnitudes are both equal to 1:
\[ |\hat{A}| = 1 \quad \text{and} \quad |\hat{B}| = 1 \]
Substituting these values into the formula gives:
\[ |\hat{A} - \hat{B}| = \sqrt{1^2 + 1^2 - 2 \cdot 1 \cdot 1 \cdot \cos(\theta)} \]

### Step 4: Simplify the Expression
Now we simplify the expression:
\[ |\hat{A} - \hat{B}| = \sqrt{1 + 1 - 2\cos(\theta)} = \sqrt{2 - 2\cos(\theta)} \]

### Step 5: Factor Out the Common Terms
We can factor out the 2 from the square root:
\[ |\hat{A} - \hat{B}| = \sqrt{2(1 - \cos(\theta))} \]

### Step 6: Use the Trigonometric Identity
Using the trigonometric identity \(1 - \cos(\theta) = 2\sin^2(\frac{\theta}{2})\), we can rewrite the expression:
\[ |\hat{A} - \hat{B}| = \sqrt{2 \cdot 2\sin^2\left(\frac{\theta}{2}\right)} = \sqrt{4\sin^2\left(\frac{\theta}{2}\right)} = 2\sin\left(\frac{\theta}{2}\right) \]

### Conclusion
Thus, the magnitude of the difference between the two unit vectors is:
\[ |\hat{A} - \hat{B}| = 2\sin\left(\frac{\theta}{2}\right) \]

### Final Answer
The value of \(|\hat{A} - \hat{B}|\) is \(2\sin\left(\frac{\theta}{2}\right)\).

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