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```python
# AI Dev Tool: PostgreSQL Vector Search Extension with FastAPI and Hono
# =====================================================================

from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from fastapi.requests import Request
from hono import Hono
from typing import List, Dict
import psycopg2
import psycopg2.extras
from psycopg2 import Error

# Initialize FastAPI app and Hono
app = FastAPI()
hono = Hono()

# PostgreSQL connection settings
POSTGRES_HOST = "localhost"
POSTGRES_DB = "vector_search"
POSTGRES_USER = "postgres"
POSTGRES_PASSWORD = "postgres"

# Create a PostgreSQL connection pool
def get_postgres_connection():
    """Returns a PostgreSQL connection object."""
    try:
        return psycopg2.connect(
            host=POSTGRES_HOST,
            database=POSTGRES_DB,
            user=POSTGRES_USER,
            password=POSTGRES_PASSWORD
        )
    except Error as e:
        print(f"Error connecting to PostgreSQL: {e}")
        raise HTTPException(status_code=500, detail="PostgreSQL connection failed")

# Define the Vector Search Extension API endpoint
@app.post("/vector_search/")
async def vector_search(request: Request):
    """
    Performs a vector search

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