To find the shortest path from vertex $E$ to vertex $F$ in the given graph, we can apply Dijkstra's algorithm or simply inspect the paths manually since the graph is small.

Step 1: List all possible paths from $E$ to $F$:

1. $E \rightarrow A \rightarrow C \rightarrow F$
2. $E \rightarrow A \rightarrow D \rightarrow F$
3. $E \rightarrow B \rightarrow D \rightarrow F$
4. $E \rightarrow B \rightarrow C \rightarrow F$
5. $E \rightarrow D \rightarrow F$

Step 2: Calculate the total weights for each path:

1. $E \rightarrow A \rightarrow C \rightarrow F: 3 + 6 + 6 = 15$
2. $E \rightarrow A \rightarrow D \rightarrow F: 3 + 1 + 3 = 7$
3. $E \rightarrow B \rightarrow D \rightarrow F: 7 + 4 + 3 = 14$
4. $E \rightarrow B \rightarrow C \rightarrow F: 7 + 1 + 6 = 14$
5. $E \rightarrow D \rightarrow F: 7 + 3 = 10$

Step 3: Identify the shortest path:

• The shortest path is $E \rightarrow A \rightarrow D \rightarrow F$ with a total weight of 7.

Thus, the shortest path from $E$ to $F$ is $E \rightarrow A \rightarrow D \rightarrow F$ with a distance of 7.

Would you like to explore more details or have any further questions?

Here are some related questions:

1. What is Dijkstra's algorithm, and how does it work?
2. How would the shortest path change if the weight between $E$ and $B$ was reduced?
3. Can you explain why some paths are longer than others?
4. How does this problem relate to real-world applications of shortest path algorithms?
5. What if we were looking for the longest path instead of the shortest?

Tip: When solving shortest path problems, ensure you evaluate all possible paths to avoid missing the optimal solution.