If `hat(a) and hat(b)` are unit vectors such that `(hat(a) + hat(b))` is a unit vector, what is the angle between `hat(a) and hat(b)`?

Explanation

To find the angle between the unit vectors \(\hat{a}\) and \(\hat{b}\) given that \((\hat{a} + \hat{b})\) is also a unit vector, we can follow these steps:

### Step 1: Understand the Magnitudes
Since \(\hat{a}\) and \(\hat{b}\) are unit vectors, we have:
\[ |\hat{a}| = 1 \quad \text{and} \quad |\hat{b}| = 1 \]
Also, the magnitude of their sum is given as:
\[ |\hat{a} + \hat{b}| = 1 \]

### Step 2: Use the Magnitude Formula
The magnitude of the sum of two vectors can be expressed as:
\[ |\hat{a} + \hat{b}|^2 = |\hat{a}|^2 + |\hat{b}|^2 + 2(\hat{a} \cdot \hat{b}) \]
Substituting the known magnitudes:
\[ 1^2 = 1^2 + 1^2 + 2(\hat{a} \cdot \hat{b}) \]
This simplifies to:
\[ 1 = 1 + 1 + 2(\hat{a} \cdot \hat{b}) \]

### Step 3: Simplify the Equation
Rearranging the equation gives:
\[ 1 = 2 + 2(\hat{a} \cdot \hat{b}) \]
Subtracting 2 from both sides:
\[ -1 = 2(\hat{a} \cdot \hat{b}) \]
Dividing both sides by 2:
\[ \hat{a} \cdot \hat{b} = -\frac{1}{2} \]

### Step 4: Relate Dot Product to Angle
The dot product of two unit vectors is also given by:
\[ \hat{a} \cdot \hat{b} = |\hat{a}||\hat{b}|\cos(\theta) \]
Since both vectors are unit vectors:
\[ \hat{a} \cdot \hat{b} = 1 \cdot 1 \cdot \cos(\theta) = \cos(\theta) \]
Thus, we have:
\[ \cos(\theta) = -\frac{1}{2} \]

### Step 5: Find the Angle
The angle \(\theta\) for which \(\cos(\theta) = -\frac{1}{2}\) corresponds to:
\[ \theta = 120^\circ \quad \text{or} \quad \theta = \frac{2\pi}{3} \text{ radians} \]

### Final Answer
The angle between \(\hat{a}\) and \(\hat{b}\) is:
\[ \theta = 120^\circ \]

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