If `veca` and `vecb` are two unit vectors such that `veca+2vecb` and `5veca-4vecb` are perpendicular to each other, then the angle between `veca` and `vecb` is

Options

A. `45^(@)`
B. `60^(@)`
C. `cos^(-1)""((1)/(3))`
D. `cos^(-1)""((2)/(7))`

Answer

To solve the problem, we need to find the angle between two unit vectors \( \vec{a} \) and \( \vec{b} \) given that the vectors \( \vec{a} + 2\vec{b} \) and \( 5\vec{a} - 4\vec{b} \) are perpendicular to each other.

### Step-by-Step Solution:

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

2. **Expand the Dot Product:**
Using the distributive property of the dot product, we expand the left-hand side:
\[ \vec{a} \cdot (5\vec{a}) + \vec{a} \cdot (-4\vec{b}) + 2\vec{b} \cdot (5\vec{a}) + 2\vec{b} \cdot (-4\vec{b}) = 0 \]
This simplifies to:
\[ 5(\vec{a} \cdot \vec{a}) - 4(\vec{a} \cdot \vec{b}) + 10(\vec{b} \cdot \vec{a}) - 8(\vec{b} \cdot \vec{b}) = 0 \]

3. **Substitute Values for Unit Vectors:**
Since \( \vec{a} \) and \( \vec{b} \) are unit vectors, we have:
\[ \vec{a} \cdot \vec{a} = 1 \quad \text{and} \quad \vec{b} \cdot \vec{b} = 1 \]
Thus, we can substitute:
\[ 5(1) - 4(\vec{a} \cdot \vec{b}) + 10(\vec{a} \cdot \vec{b}) - 8(1) = 0 \]
This simplifies to:
\[ 5 - 8 + 6(\vec{a} \cdot \vec{b}) = 0 \]

4. **Simplify the Equation:**
Rearranging gives:
\[ 6(\vec{a} \cdot \vec{b}) = 3 \]
Therefore:
\[ \vec{a} \cdot \vec{b} = \frac{3}{6} = \frac{1}{2} \]

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

6. **Find the Angle:**
The angle \( \theta \) whose cosine is \( \frac{1}{2} \) is:
\[ \theta = 60^\circ \]

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