Let ` hat a` and ` hat b` be two unit vectors. If the vectors ` vec c= hat a+2 hat b` and ` vec d=5 hat a-4 hat b` are perpendicular to each other, then the angle between ` hat a` and ` hat b` is

Options

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

Explanation

To solve the problem, we need to find 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 Perpendicular Vectors**:
Two vectors are perpendicular if their dot product is zero. Therefore, we need to compute the dot product of \(\vec{c}\) and \(\vec{d}\) and set it to zero:
\[ \vec{c} \cdot \vec{d} = 0 \]

2. **Calculate \(\vec{c} \cdot \vec{d}\)**:
Substitute \(\vec{c}\) and \(\vec{d}\):
\[ \vec{c} = \hat{a} + 2\hat{b}, \quad \vec{d} = 5\hat{a} - 4\hat{b} \]
Now calculate the dot product:
\[ \vec{c} \cdot \vec{d} = (\hat{a} + 2\hat{b}) \cdot (5\hat{a} - 4\hat{b}) \]

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

4. **Substituting Unit Vector Properties**:
Since \(\hat{a}\) and \(\hat{b}\) are unit vectors:
\[ \hat{a} \cdot \hat{a} = 1, \quad \hat{b} \cdot \hat{b} = 1 \]
Thus, we have:
\[ \vec{c} \cdot \vec{d} = 5(1) - 4(\hat{a} \cdot \hat{b}) + 10(\hat{a} \cdot \hat{b}) - 8(1) \]
Simplifying gives:
\[ = 5 - 8 + 6(\hat{a} \cdot \hat{b}) = -3 + 6(\hat{a} \cdot \hat{b}) \]

5. **Setting the Dot Product to Zero**:
Since \(\vec{c}\) and \(\vec{d}\) are perpendicular:
\[ -3 + 6(\hat{a} \cdot \hat{b}) = 0 \]
Rearranging gives:
\[ 6(\hat{a} \cdot \hat{b}) = 3 \implies \hat{a} \cdot \hat{b} = \frac{1}{2} \]

6. **Finding the Angle**:
The dot product of two unit vectors is 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:
\[ \cos \theta = \frac{1}{2} \]

7. **Calculating the Angle**:
The angle \(\theta\) corresponding to \(\cos \theta = \frac{1}{2}\) is:
\[ \theta = \frac{\pi}{3} \text{ or } 60^\circ \]

### Final Answer:
The angle between the unit vectors \(\hat{a}\) and \(\hat{b}\) is \(\frac{\pi}{3}\) radians or \(60^\circ\).