Let A, B, C, D, E represent vertices of a regular pentagon ABCDE. Given the position vector of these vertices be `veca, veca + vecb, vecb, lamda veca and lamda vecb`, respectively.
AD divides EC in the ratio

Options

A. `cos""(2pi)/(5) : 1`
B. `cos""(3pi)/(5) : 1`
C. `1 : 2 cos""(pi)/(5)`
D. `1 : 2`

Explanation

To solve the problem of finding the ratio in which line segment AD divides line segment EC in a regular pentagon ABCDE, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Pentagon Structure**:
- A regular pentagon has equal sides and equal angles. The vertices are labeled as A, B, C, D, and E.

2. **Identify the Position Vectors**:
- The position vectors of the vertices are given as follows:
- \( \vec{A} = \vec{a} \)
- \( \vec{B} = \vec{a} + \vec{b} \)
- \( \vec{C} = \vec{b} \)
- \( \vec{D} = \lambda \vec{a} \)
- \( \vec{E} = \lambda \vec{b} \)

3. **Visualize the Problem**:
- Draw the pentagon and label the vertices A, B, C, D, and E. Draw the diagonals AD and EC.

4. **Use the Properties of the Pentagon**:
- In a regular pentagon, the diagonals intersect at specific angles. The angles between the diagonals can be used to find the ratios of the segments.

5. **Apply the Law of Sines**:
- In triangle OAB (formed by the origin O and points A and B), we can use the Law of Sines to find the lengths of the segments.
- Let \( O \) be the center of the pentagon. The angles at O can be calculated based on the regular pentagon's properties.

6. **Calculate the Lengths**:
- Assign lengths to the sides of the triangle based on the properties of the pentagon. For instance, let \( OC = CD = 1 \) (for simplicity).
- Using the Law of Sines:
\[ \frac{OD}{\sin(\text{angle opposite to } OD)} = \frac{OC}{\sin(\text{angle opposite to } OC)} \]

7. **Determine the Ratio**:
- From the calculations, we find the lengths of segments OD and OC.
- The ratio \( \frac{AD}{EC} \) can then be expressed in terms of the lengths found.

8. **Final Calculation**:
- Substitute the values obtained from the Law of Sines into the ratio to find the exact ratio in which AD divides EC.

9. **Conclusion**:
- The final answer will be in the form of \( \frac{1}{2 \cos(\frac{5\pi}{5})} \).