If vector `(hata+ 2hatb)` is perpendicular to vector `(5hata-4hatb)`, then find the angle between `hata and hatb`.

Explanation

To solve the problem, we need to find the angle between the unit vectors \( \hat{a} \) and \( \hat{b} \) given that the vector \( \hat{a} + 2\hat{b} \) is perpendicular to the vector \( 5\hat{a} - 4\hat{b} \).

### Step-by-Step Solution:

1. **Understanding Perpendicular Vectors**:
Two vectors are perpendicular if their dot product is zero. Therefore, we can write the equation:
\[ (\hat{a} + 2\hat{b}) \cdot (5\hat{a} - 4\hat{b}) = 0 \]

2. **Expanding the Dot Product**:
Now, we expand the dot product:
\[ \hat{a} \cdot (5\hat{a}) + \hat{a} \cdot (-4\hat{b}) + 2\hat{b} \cdot (5\hat{a}) + 2\hat{b} \cdot (-4\hat{b}) \]
This simplifies to:
\[ 5(\hat{a} \cdot \hat{a}) - 4(\hat{a} \cdot \hat{b}) + 10(\hat{b} \cdot \hat{a}) - 8(\hat{b} \cdot \hat{b}) \]

3. **Using Properties of Dot Products**:
Since \( \hat{a} \) and \( \hat{b} \) are unit vectors, we know:
\[ \hat{a} \cdot \hat{a} = 1 \quad \text{and} \quad \hat{b} \cdot \hat{b} = 1 \]
Also, \( \hat{a} \cdot \hat{b} = \hat{b} \cdot \hat{a} \). Thus, we can rewrite the equation:
\[ 5(1) - 4(\hat{a} \cdot \hat{b}) + 10(\hat{a} \cdot \hat{b}) - 8(1) = 0 \]
This simplifies to:
\[ 5 - 8 + 6(\hat{a} \cdot \hat{b}) = 0 \]

4. **Solving for the Dot Product**:
Combine the constants:
\[ -3 + 6(\hat{a} \cdot \hat{b}) = 0 \]
Rearranging gives:
\[ 6(\hat{a} \cdot \hat{b}) = 3 \]
Therefore:
\[ \hat{a} \cdot \hat{b} = \frac{1}{2} \]

5. **Finding the Angle**:
The dot product of two vectors can also be expressed in terms of the angle \( \theta \) between them:
\[ \hat{a} \cdot \hat{b} = |\hat{a}| |\hat{b}| \cos(\theta) \]
Since both \( \hat{a} \) and \( \hat{b} \) are unit vectors, their magnitudes are 1:
\[ \hat{a} \cdot \hat{b} = 1 \cdot 1 \cdot \cos(\theta) = \cos(\theta) \]
Thus, we have:
\[ \cos(\theta) = \frac{1}{2} \]

6. **Calculating the Angle**:
The angle \( \theta \) corresponding to \( \cos(\theta) = \frac{1}{2} \) is:
\[ \theta = 60^\circ \]

### Final Answer:
The angle between \( \hat{a} \) and \( \hat{b} \) is \( 60^\circ \).