Two unit vector when added give a unit vector . Then choose the correct statement.

Options

A. magnitude the their difference is `sqrt(3)`
B. Mignitude of their difference is 1
C. Angle between the vectors is `90^(@)`
D. Angle between the sum and the difference of the two vectors is `90^(@)`

Explanation

To solve the problem step by step, we will analyze the given information about the two unit vectors and their resultant.

### Step 1: Understand the Problem
We have two unit vectors, let's denote them as **A** and **B**. When these vectors are added, they yield another unit vector **R**. We need to find the correct statement regarding these vectors.

### Step 2: Write the Magnitude of the Resultant Vector
The magnitude of the resultant vector **R** can be expressed using the formula:
\[ |R| = \sqrt{|A|^2 + |B|^2 + 2|A||B|\cos\theta} \]
Since both **A** and **B** are unit vectors, their magnitudes are 1:
\[ |A| = |B| = 1 \]
Thus, the equation simplifies to:
\[ |R| = \sqrt{1^2 + 1^2 + 2 \cdot 1 \cdot 1 \cdot \cos\theta} = \sqrt{2 + 2\cos\theta} \]

### Step 3: Set the Magnitude of the Resultant Equal to 1
According to the problem, the resultant vector **R** is also a unit vector, so:
\[ |R| = 1 \]
Setting the two expressions for the magnitude equal gives:
\[ 1 = \sqrt{2 + 2\cos\theta} \]

### Step 4: Square Both Sides
Squaring both sides to eliminate the square root:
\[ 1^2 = (2 + 2\cos\theta) \]
This simplifies to:
\[ 1 = 2 + 2\cos\theta \]

### Step 5: Solve for Cosine
Rearranging the equation gives:
\[ 2\cos\theta = 1 - 2 \]
\[ 2\cos\theta = -1 \]
\[ \cos\theta = -\frac{1}{2} \]

### Step 6: Find the Angle
The angle whose cosine is \(-\frac{1}{2}\) is:
\[ \theta = 120^\circ \]

### Step 7: Find the Magnitude of the Difference of the Vectors
The difference of the two vectors **A** and **B** can be expressed as:
\[ |A - B| = \sqrt{|A|^2 + |B|^2 - 2|A||B|\cos\theta} \]
Substituting the values:
\[ |A - B| = \sqrt{1^2 + 1^2 - 2 \cdot 1 \cdot 1 \cdot \cos(120^\circ)} \]
Since \(\cos(120^\circ) = -\frac{1}{2}\):
\[ |A - B| = \sqrt{1 + 1 + 1} = \sqrt{3} \]

### Step 8: Analyze the Statements
We need to check the statements provided in the question. Based on our calculations:
1. The angle between the two vectors is \(120^\circ\).
2. The magnitude of the difference of the vectors is \(\sqrt{3}\).
3. The angle between the sum and the difference of the vectors can be shown to be \(90^\circ\).

### Conclusion
The correct statements based on the calculations are:
- The angle between the two unit vectors is \(120^\circ\).
- The magnitude of the difference of the vectors is \(\sqrt{3}\).
- The angle between the sum and the difference of the vectors is \(90^\circ\).