Let `hata and hatb` be two unit vectors. If the vectors `vecc=hata+2hatb and vecd=5hata-4hatb` are perpendicular to each other then the angle between `hata and hatb` is (A) `pi/2` (B) `pi/3` (C) `pi/4` (D) `pi/6`

Options

A. `pi/6`
B. `pi/2`
C. `pi/3`
D. `pi/4`

Answer

To solve the problem, we need to determine the angle between the unit vectors \(\hat{a}\) and \(\hat{b}\) given that the vectors \(\vec{c} = \hat{a} + 2\hat{b}\) and \(\vec{d} = 5\hat{a} - 4\hat{b}\) are perpendicular.

### Step-by-Step Solution:

1. **Understanding the Condition of Perpendicularity**:
For two vectors to be perpendicular, their dot product must equal zero. Therefore, we have:
\[ \vec{c} \cdot \vec{d} = 0 \]

2. **Substituting the Expressions for \(\vec{c}\) and \(\vec{d}\)**:
Substitute \(\vec{c} = \hat{a} + 2\hat{b}\) and \(\vec{d} = 5\hat{a} - 4\hat{b}\):
\[ (\hat{a} + 2\hat{b}) \cdot (5\hat{a} - 4\hat{b}) = 0 \]

3. **Expanding the Dot Product**:
Using the distributive property of the dot product:
\[ \hat{a} \cdot (5\hat{a}) + \hat{a} \cdot (-4\hat{b}) + 2\hat{b} \cdot (5\hat{a}) + 2\hat{b} \cdot (-4\hat{b}) = 0 \]
This simplifies to:
\[ 5(\hat{a} \cdot \hat{a}) - 4(\hat{a} \cdot \hat{b}) + 10(\hat{b} \cdot \hat{a}) - 8(\hat{b} \cdot \hat{b}) = 0 \]

4. **Using the Properties of Unit Vectors**:
Since \(\hat{a}\) and \(\hat{b}\) are unit vectors, we have:
\[ \hat{a} \cdot \hat{a} = 1 \quad \text{and} \quad \hat{b} \cdot \hat{b} = 1 \]
Thus, the equation becomes:
\[ 5 - 4(\hat{a} \cdot \hat{b}) + 10(\hat{a} \cdot \hat{b}) - 8 = 0 \]
Simplifying further:
\[ -3 + 6(\hat{a} \cdot \hat{b}) = 0 \]

5. **Solving for \(\hat{a} \cdot \hat{b}\)**:
Rearranging gives:
\[ 6(\hat{a} \cdot \hat{b}) = 3 \implies \hat{a} \cdot \hat{b} = \frac{1}{2} \]

6. **Finding the Angle Between \(\hat{a}\) and \(\hat{b}\)**:
The dot product of two vectors is also given by:
\[ \hat{a} \cdot \hat{b} = |\hat{a}| |\hat{b}| \cos \theta \]
Since both are unit vectors, this simplifies to:
\[ \hat{a} \cdot \hat{b} = \cos \theta \]
Therefore, we have:
\[ \cos \theta = \frac{1}{2} \]

7. **Determining the Angle \(\theta\)**:
The angle \(\theta\) corresponding to \(\cos \theta = \frac{1}{2}\) is:
\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

### Final Answer:
The angle between \(\hat{a}\) and \(\hat{b}\) is \(\frac{\pi}{3}\).