If `vec(e_(1))` and `vec(e_(2))` are two unit vectors and `theta` is the angle between them, then `sin (theta/2)` is:

Options

A. `1/2 |vec(e)_(1)+vec(e)_(2)|`
B. `1/2 |vec(e)_(1)-vec(e)_(2)|`
C. `(vec(e)_(1).vec(e)_(2))/(2)`
D. `(|vec(e)_(1)xxvec(e)_(2)|)/(2|vec(e)_(1)||vec(e)_(2)|)`

Explanation

To find the value of \( \sin(\theta/2) \) where \( \vec{e_1} \) and \( \vec{e_2} \) are two unit vectors and \( \theta \) is the angle between them, we can follow these steps:

### Step 1: Understand the given information
We know that \( \vec{e_1} \) and \( \vec{e_2} \) are unit vectors. This means:
\[ |\vec{e_1}| = 1 \quad \text{and} \quad |\vec{e_2}| = 1 \]
The angle between these two vectors is given as \( \theta \).

### Step 2: Use the formula for the magnitude of the difference of two vectors
The magnitude of the difference of two vectors can be expressed as:
\[ |\vec{e_1} - \vec{e_2}| = \sqrt{|\vec{e_1}|^2 + |\vec{e_2}|^2 - 2|\vec{e_1}||\vec{e_2}|\cos(\theta)} \]
Substituting the values for unit vectors:
\[ |\vec{e_1} - \vec{e_2}| = \sqrt{1^2 + 1^2 - 2 \cdot 1 \cdot 1 \cdot \cos(\theta)} \]
This simplifies to:
\[ |\vec{e_1} - \vec{e_2}| = \sqrt{2 - 2\cos(\theta)} \]

### Step 3: Factor out the common terms
We can factor out the 2 from the square root:
\[ |\vec{e_1} - \vec{e_2}| = \sqrt{2(1 - \cos(\theta))} \]

### Step 4: Use the identity for \( 1 - \cos(\theta) \)
We know from trigonometric identities that:
\[ 1 - \cos(\theta) = 2\sin^2\left(\frac{\theta}{2}\right) \]
Substituting this into our equation gives:
\[ |\vec{e_1} - \vec{e_2}| = \sqrt{2 \cdot 2\sin^2\left(\frac{\theta}{2}\right)} = \sqrt{4\sin^2\left(\frac{\theta}{2}\right)} \]

### Step 5: Simplify the expression
Taking the square root results in:
\[ |\vec{e_1} - \vec{e_2}| = 2\sin\left(\frac{\theta}{2}\right) \]

### Step 6: Solve for \( \sin\left(\frac{\theta}{2}\right) \)
From the equation above, we can express \( \sin\left(\frac{\theta}{2}\right) \):
\[ \sin\left(\frac{\theta}{2}\right) = \frac{1}{2} |\vec{e_1} - \vec{e_2}| \]

### Conclusion
Thus, the final expression for \( \sin\left(\frac{\theta}{2}\right) \) is:
\[ \sin\left(\frac{\theta}{2}\right) = \frac{1}{2} |\vec{e_1} - \vec{e_2}| \]