To solve this problem, we'll apply a graph-based clustering algorithm. Specifically, we’ll use a threshold-based approach, where two points (vectors) are connected if the Euclidean distance between them is less than or equal to 2.5. From these connections, we can derive the clusters.

Steps:

1. Compute the Euclidean distance between every pair of vectors.
2. Create a graph where each point is a node, and edges exist between nodes if the distance between them is ≤ 2.5.
3. Identify the connected components, which will give us the clusters.

Euclidean Distance Formula:

$d((x_1, y_1), (x_2, y_2)) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Coordinates of the Vectors:

• A = (1, 1)
• B = (1, 2)
• C = (3, 2)
• D = (5, 4)
• E = (5, 8)
• F = (6, 9)

1. Compute pairwise distances:

• $d(A, B) = \sqrt{(1 - 1)^2 + (2 - 1)^2} = 1$

• $d(A, C) = \sqrt{(3 - 1)^2 + (2 - 1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236$

• $d(A, D) = \sqrt{(5 - 1)^2 + (4 - 1)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

• $d(A, E) = \sqrt{(5 - 1)^2 + (8 - 1)^2} = \sqrt{16 + 49} = \sqrt{65} \approx 8.062$

• $d(A, F) = \sqrt{(6 - 1)^2 + (9 - 1)^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.434$

• $d(B, C) = \sqrt{(3 - 1)^2 + (2 - 2)^2} = \sqrt{4 + 0} = \sqrt{4} = 2$

• $d(B, D) = \sqrt{(5 - 1)^2 + (4 - 2)^2} = \sqrt{16 + 4} = \sqrt{20} \approx 4.472$

• $d(B, E) = \sqrt{(5 - 1)^2 + (8 - 2)^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.211$

• $d(B, F) = \sqrt{(6 - 1)^2 + (9 - 2)^2} = \sqrt{25 + 49} = \sqrt{74} \approx 8.602$

• $d(C, D) = \sqrt{(5 - 3)^2 + (4 - 2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.828$

• $d(C, E) = \sqrt{(5 - 3)^2 + (8 - 2)^2} = \sqrt{4 + 36} = \sqrt{40} \approx 6.325$

• $d(C, F) = \sqrt{(6 - 3)^2 + (9 - 2)^2} = \sqrt{9 + 49} = \sqrt{58} \approx 7.616$

• $d(D, E) = \sqrt{(5 - 5)^2 + (8 - 4)^2} = \sqrt{0 + 16} = 4$

• $d(D, F) = \sqrt{(6 - 5)^2 + (9 - 4)^2} = \sqrt{1 + 25} = \sqrt{26} \approx 5.099$

• $d(E, F) = \sqrt{(6 - 5)^2 + (9 - 8)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.414$

2. Graph Representation:

We connect vectors if their distance is ≤ 2.5:

• $A$ is connected to $B$ (distance = 1).
• $A$ is connected to $C$ (distance ≈ 2.236).
• $B$ is connected to $C$ (distance = 2).
• $E$ is connected to $F$ (distance ≈ 1.414).

3. Clusters:

Based on the graph, we have two connected components:

• Component 1: $A, B, C$ (since they are all connected by distances ≤ 2.5)
• Component 2: $E, F$ (since $E$ and $F$ are connected, but not to the others)
• $D$ remains disconnected, as it has no distances ≤ 2.5 to any other point.

4. Identify the Correct Clustering Option:

• (A, B, C) forms one cluster.
• (E, F) forms another cluster.
• D is its own cluster.

The correct answer is: b) (A, B, C), (D), (E, F).

---

Would you like any further clarification or details on this process?

5 Related Questions:

1. How does the Euclidean distance threshold affect the number of clusters in a graph-based algorithm?
2. What are the benefits of using graph-based clustering over k-means for non-spherical data?
3. Can graph-based algorithms be used with different distance metrics (e.g., Manhattan distance)?
4. How does changing the threshold influence the formation of connected components in the graph?
5. What are common real-world applications of graph-based clustering algorithms?

Tip: In graph-based clustering, adjusting the distance threshold can greatly influence the granularity of the clusters, from many small clusters to a few large ones.