Introduction:
The problem states that the sum of two unit vectors is also a unit vector and we need to find the magnitude of their difference. In this answer, we will explain how to solve this problem step by step.

Understand the problem:
We have two unit vectors, let's say A and B, and their sum is also a unit vector. We need to find the magnitude of their difference, which is |A-B|.

Use the given information:
Since A and B are unit vectors, their magnitudes are 1. Also, the sum of A and B is also a unit vector, so its magnitude is also 1.

Apply vector addition formula:
We know that the magnitude of the sum of two vectors is given by the formula: |A + B| = sqrt(A^2 + B^2 + 2ABcosθ), where θ is the angle between vectors A and B.

Simplify the formula:
Since A and B are unit vectors, their magnitudes are 1. Therefore, the formula simplifies to: |A + B| = sqrt(2 + 2cosθ).

Since the sum of A and B is a unit vector, its magnitude is also 1. Therefore, we can write: sqrt(2 + 2cosθ) = 1.

Solving for cosθ, we get: cosθ = -1/2.

Find the angle between A and B:
Since cosθ = -1/2, the angle θ between A and B is 120 degrees.

Apply vector subtraction formula:
We know that the magnitude of the difference between two vectors is given by the formula: |A - B| = sqrt(A^2 + B^2 - 2ABcosθ).

Simplify the formula:
Substituting the values of A, B, and cosθ, we get:

|A - B| = sqrt(1 + 1 - 2cos120) = sqrt(3).

Therefore, the magnitude of the difference between A and B is sqrt(3).

Conclusion:
In this answer, we explained how to find the magnitude of the difference between two unit vectors whose sum is also a unit vector. We used vector addition and subtraction formulas to simplify the problem and obtained the final answer of sqrt(3).

√3