If the sum of two unit vectors is a unit vector, prove that the magnitude of their difference is `sqrt(3.)`

Explanation

To solve the problem, we need to prove that if the sum of two unit vectors \( \hat{A} \) and \( \hat{B} \) is also a unit vector, then the magnitude of their difference \( |\hat{A} - \hat{B}| \) is \( \sqrt{3} \).

### Step-by-Step Solution:

1. **Define the Unit Vectors**:
Let \( \hat{A} \) and \( \hat{B} \) be two unit vectors. By definition, this means:
\[
|\hat{A}| = 1 \quad \text{and} \quad |\hat{B}| = 1
...