The difference of two unit vectors is found to have a magnitude of unity . The angle between them is

Answer

To solve the problem step by step, we will use the properties of vectors and the information given in the question.

### Step 1: Define the unit vectors
Let the two unit vectors be represented as \( \hat{a} \) and \( \hat{b} \).

### Step 2: Write the equation for the magnitude of their difference
We know that the magnitude of the difference of the two unit vectors is given as:
\[ |\hat{a} - \hat{b}| = 1 \]

### Step 3: Square both sides of the equation
To eliminate the square root, we square both sides:
\[ |\hat{a} - \hat{b}|^2 = 1^2 \]
This gives us:
\[ |\hat{a} - \hat{b}|^2 = 1 \]

### Step 4: Expand the left-hand side using the formula for the magnitude of a vector
Using the formula for the magnitude of the difference of two vectors, we have:
\[ |\hat{a} - \hat{b}|^2 = |\hat{a}|^2 + |\hat{b}|^2 - 2 |\hat{a}| |\hat{b}| \cos(\theta) \]
where \( \theta \) is the angle between the two vectors.

### Step 5: Substitute the magnitudes of the unit vectors
Since both \( \hat{a} \) and \( \hat{b} \) are unit vectors, their magnitudes are 1:
\[ |\hat{a}|^2 = 1 \quad \text{and} \quad |\hat{b}|^2 = 1 \]
Substituting these values into the equation gives:
\[ 1 + 1 - 2 \cdot 1 \cdot 1 \cdot \cos(\theta) = 1 \]
This simplifies to:
\[ 2 - 2 \cos(\theta) = 1 \]

### Step 6: Rearrange the equation
Rearranging the equation, we have:
\[ 2 - 1 = 2 \cos(\theta) \]
which simplifies to:
\[ 1 = 2 \cos(\theta) \]

### Step 7: Solve for \( \cos(\theta) \)
Dividing both sides by 2 gives:
\[ \cos(\theta) = \frac{1}{2} \]

### Step 8: Determine the angle \( \theta \)
The angle \( \theta \) for which \( \cos(\theta) = \frac{1}{2} \) is:
\[ \theta = 60^\circ \quad \text{or} \quad \theta = \frac{\pi}{3} \text{ radians} \]

### Final Answer
Thus, the angle between the two unit vectors is \( 60^\circ \) or \( \frac{\pi}{3} \) radians.

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