If `veca` and `vecb` are unit vectors inclined at an angle `theta`, then the value of `|veca-vecb|` is

Options

A. `2 sin(theta)/(2)`
B. `2sintheta`
C. `2costheta(theta)/(2)`
D. `2costheta`

Answer

To find the value of \( |\vec{a} - \vec{b}| \) where \( \vec{a} \) and \( \vec{b} \) are unit vectors inclined at an angle \( \theta \), we can follow these steps:

### Step 1: Write the expression for the magnitude
We start with the expression for the magnitude:
\[ |\vec{a} - \vec{b}| \]

### Step 2: Square the magnitude
To simplify the calculation, we square the magnitude:
\[ |\vec{a} - \vec{b}|^2 = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b}) \]

### Step 3: Expand the dot product
Using the properties of the dot product, we expand the expression:
\[ |\vec{a} - \vec{b}|^2 = \vec{a} \cdot \vec{a} - 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} \]

### Step 4: Substitute the values for unit vectors
Since \( \vec{a} \) and \( \vec{b} \) are unit vectors, we have:
\[ \vec{a} \cdot \vec{a} = 1 \quad \text{and} \quad \vec{b} \cdot \vec{b} = 1 \]
Thus, we can substitute these values:
\[ |\vec{a} - \vec{b}|^2 = 1 - 2 \vec{a} \cdot \vec{b} + 1 \]
This simplifies to:
\[ |\vec{a} - \vec{b}|^2 = 2 - 2 \vec{a} \cdot \vec{b} \]

### Step 5: Use the dot product formula
The dot product of two unit vectors can be expressed in terms of the angle \( \theta \) between them:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) = 1 \cdot 1 \cdot \cos(\theta) = \cos(\theta) \]
Substituting this into our equation gives:
\[ |\vec{a} - \vec{b}|^2 = 2 - 2 \cos(\theta) \]

### Step 6: Factor out the common term
We can factor out the 2:
\[ |\vec{a} - \vec{b}|^2 = 2(1 - \cos(\theta)) \]

### Step 7: Apply the half-angle identity
Using the trigonometric identity \( 1 - \cos(\theta) = 2 \sin^2\left(\frac{\theta}{2}\right) \), we can rewrite the equation:
\[ |\vec{a} - \vec{b}|^2 = 2 \cdot 2 \sin^2\left(\frac{\theta}{2}\right) = 4 \sin^2\left(\frac{\theta}{2}\right) \]

### Step 8: Take the square root
Finally, we take the square root to find the magnitude:
\[ |\vec{a} - \vec{b}| = \sqrt{4 \sin^2\left(\frac{\theta}{2}\right)} = 2 \sin\left(\frac{\theta}{2}\right) \]

### Final Answer
Thus, the value of \( |\vec{a} - \vec{b}| \) is:
\[ |\vec{a} - \vec{b}| = 2 \sin\left(\frac{\theta}{2}\right) \]