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```python
# AI Dev Tool: PostgreSQL Vector Search Extension with FastAPI and Supabase
# ==============================================================================
# This is a production-ready developer code asset that integrates PostgreSQL Vector Search Extension
# with FastAPI and Supabase for building scalable and secure micro-SaaS applications.

from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from fastapi.requests import Request
from supabase import create_client, Client
from pydantic import BaseModel
from typing import List, Optional
import os
import logging

# Initialize the logger
logging.basicConfig(level=logging.INFO)
logger = logging.getLogger(__name__)

# Initialize the FastAPI app
app = FastAPI()

# Initialize the Supabase client
SUPABASE_URL = os.environ.get("SUPABASE_URL")
SUPABASE_KEY = os.environ.get("SUPABASE_KEY")
supabase: Client = create_client(SUPABASE_URL, SUPABASE_KEY)

# Define the PostgreSQL Vector Search Extension model
class VectorSearchModel(BaseModel):
    id: int
    vector: List[float]
    metadata: Optional[dict]

# Define the API endpoint for vector search
@app.post("/vector_search")
async def vector_search(request: Request, query: List[float]):
    """

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