If the sum of two unit vectors is a unit vector, then the magnitude of their difference is :

Options

A. `sqrt(3)`
B. `sqrt(2)`
C. `sqrt(5)`
D. `(1)/(sqrt(2))`

Explanation

To solve the problem, we need to find the magnitude of the difference between two unit vectors \( \mathbf{a} \) and \( \mathbf{b} \), given that their sum is also a unit vector.

### Step-by-Step Solution:

1. **Define the Unit Vectors**:
Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors. By definition, the magnitudes of both vectors are:
\[ |\mathbf{a}| = 1 \quad \text{and} \quad |\mathbf{b}| = 1 \]

2. **Sum of the Vectors**:
We know that the sum of these two vectors is also a unit vector:
\[ |\mathbf{a} + \mathbf{b}| = 1 \]
This can be expressed using the formula for the magnitude of the sum of two vectors:
\[ |\mathbf{a} + \mathbf{b}|^2 = |\mathbf{a}|^2 + |\mathbf{b}|^2 + 2(\mathbf{a} \cdot \mathbf{b}) \]
Substituting the magnitudes:
\[ 1^2 = 1^2 + 1^2 + 2(\mathbf{a} \cdot \mathbf{b}) \]
Simplifying this gives:
\[ 1 = 1 + 1 + 2(\mathbf{a} \cdot \mathbf{b}) \]
\[ 1 = 2 + 2(\mathbf{a} \cdot \mathbf{b}) \]
Rearranging gives:
\[ 2(\mathbf{a} \cdot \mathbf{b}) = 1 - 2 \]
\[ 2(\mathbf{a} \cdot \mathbf{b}) = -1 \]
Therefore,
\[ \mathbf{a} \cdot \mathbf{b} = -\frac{1}{2} \]

3. **Magnitude of the Difference**:
Now, we need to find the magnitude of the difference \( |\mathbf{a} - \mathbf{b}| \). We can use the formula:
\[ |\mathbf{a} - \mathbf{b}|^2 = |\mathbf{a}|^2 + |\mathbf{b}|^2 - 2(\mathbf{a} \cdot \mathbf{b}) \]
Substituting the known values:
\[ |\mathbf{a} - \mathbf{b}|^2 = 1^2 + 1^2 - 2\left(-\frac{1}{2}\right) \]
Simplifying this gives:
\[ |\mathbf{a} - \mathbf{b}|^2 = 1 + 1 + 1 = 3 \]
Taking the square root:
\[ |\mathbf{a} - \mathbf{b}| = \sqrt{3} \]

### Conclusion:
The magnitude of the difference between the two unit vectors is:
\[ \boxed{\sqrt{3}} \]