If `vec(a)` and `vec(b)` are unit vectors inclined at an angle `alpha`, then the value of `| vec(a) - vec(b)|` is

Options

A. `2 sin ""(alpha)/(2)`
B. `2 cos ""(alpha)/(2)`
C. `2 sin alpha`
D. `2 cos alpha`

Answer

To solve the problem, we need to find the value of \(|\vec{a} - \vec{b}|\) given that \(\vec{a}\) and \(\vec{b}\) are unit vectors inclined at an angle \(\alpha\).

### Step-by-Step Solution:

1. **Understanding the Vectors**:
- Let \(\vec{a}\) and \(\vec{b}\) be unit vectors. This means that \(|\vec{a}| = 1\) and \(|\vec{b}| = 1\).
- The angle between the two vectors is given as \(\alpha\).

2. **Using the Formula for the Magnitude of the Difference of Two Vectors**:
- We want to find \(|\vec{a} - \vec{b}|\).
- We can use the formula:
\[ |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2\vec{a} \cdot \vec{b} \]

3. **Substituting the Values**:
- Since both vectors are unit vectors:
\[ |\vec{a}|^2 = 1 \quad \text{and} \quad |\vec{b}|^2 = 1 \]
- Therefore, substituting these values into the formula gives:
\[ |\vec{a} - \vec{b}|^2 = 1 + 1 - 2\vec{a} \cdot \vec{b} \]
- This simplifies to:
\[ |\vec{a} - \vec{b}|^2 = 2 - 2\vec{a} \cdot \vec{b} \]

4. **Finding the Dot Product**:
- The dot product \(\vec{a} \cdot \vec{b}\) can be expressed in terms of the angle \(\alpha\):
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\alpha) = 1 \cdot 1 \cdot \cos(\alpha) = \cos(\alpha) \]
- Substituting this into our equation:
\[ |\vec{a} - \vec{b}|^2 = 2 - 2\cos(\alpha) \]

5. **Factoring the Expression**:
- We can factor out a 2:
\[ |\vec{a} - \vec{b}|^2 = 2(1 - \cos(\alpha)) \]

6. **Using the Trigonometric Identity**:
- We can use the identity \(1 - \cos(\alpha) = 2\sin^2\left(\frac{\alpha}{2}\right)\):
\[ |\vec{a} - \vec{b}|^2 = 2 \cdot 2\sin^2\left(\frac{\alpha}{2}\right) = 4\sin^2\left(\frac{\alpha}{2}\right) \]

7. **Taking the Square Root**:
- Finally, taking the square root to find the magnitude:
\[ |\vec{a} - \vec{b}| = \sqrt{4\sin^2\left(\frac{\alpha}{2}\right)} = 2\sin\left(\frac{\alpha}{2}\right) \]

### Final Answer:
\[ |\vec{a} - \vec{b}| = 2\sin\left(\frac{\alpha}{2}\right) \]