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

Options

A. `2 cos""(theta)/(2)`
B. `2 tan""(theta)/(2)`
C. `2 Sin""(theta)/(2)`
D. None of these

Explanation

To find the value of \(|\hat{A} - \hat{B}|\) where \(\hat{A}\) and \(\hat{B}\) are unit vectors of vectors \(A\) and \(B\) respectively, and the angle between them is \(\theta\), we can follow these steps:

### Step 1: Understand the problem
We are given two unit vectors \(\hat{A}\) and \(\hat{B}\) with an angle \(\theta\) between them. We need to find the magnitude of the difference of these two vectors, \(|\hat{A} - \hat{B}|\).

### Step 2: Use the formula for the magnitude of the difference of two vectors
The magnitude of the difference between two vectors can be expressed as:
\[ |\hat{A} - \hat{B}| = \sqrt{|\hat{A}|^2 + |\hat{B}|^2 - 2 |\hat{A}||\hat{B}|\cos(\phi)} \]
where \(\phi\) is the angle between the two vectors.

### Step 3: Substitute the values
Since \(\hat{A}\) and \(\hat{B}\) are unit vectors, we have:
\[ |\hat{A}| = 1 \quad \text{and} \quad |\hat{B}| = 1 \]
Thus, substituting these values into the formula gives:
\[ |\hat{A} - \hat{B}| = \sqrt{1^2 + 1^2 - 2 \cdot 1 \cdot 1 \cdot \cos(\theta)} \]
This simplifies to:
\[ |\hat{A} - \hat{B}| = \sqrt{1 + 1 - 2\cos(\theta)} = \sqrt{2 - 2\cos(\theta)} \]

### Step 4: Factor the expression
We can factor out the 2 from the square root:
\[ |\hat{A} - \hat{B}| = \sqrt{2(1 - \cos(\theta))} \]

### Step 5: Use the trigonometric identity
Using the trigonometric identity \(1 - \cos(\theta) = 2\sin^2(\frac{\theta}{2})\), we can rewrite the expression:
\[ |\hat{A} - \hat{B}| = \sqrt{2 \cdot 2\sin^2\left(\frac{\theta}{2}\right)} = \sqrt{4\sin^2\left(\frac{\theta}{2}\right)} = 2\sin\left(\frac{\theta}{2}\right) \]

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

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