⚡ Live Sandbox: AI Dev Tool: PostgreSQL Vector Search Extension

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```python
# AI Dev Tool: FastAPI + PostgreSQL Vector Search Extension Micro-SaaS Boilerplate
# ==============================================================================
# This is a production-ready starter boilerplate for building micro-SaaS applications
# using FastAPI, PostgreSQL, and the Vector Search Extension.

from fastapi import FastAPI, Depends
from fastapi.responses import JSONResponse
from fastapi.requests import Request
from fastapi.security import HTTPBearer, HTTPAuthorizationCredentials
from pydantic import BaseModel
from typing import List, Optional
import psycopg2
import psycopg2.extras
import json

# Initialize the FastAPI application
app = FastAPI()

# Define the security scheme for authentication
security = HTTPBearer()

# Define the PostgreSQL connection parameters
POSTGRES_HOST = "localhost"
POSTGRES_PORT = 5432
POSTGRES_USER = "postgres"
POSTGRES_PASSWORD = "postgres"
POSTGRES_DB = "mydb"

# Establish a connection to the PostgreSQL database
def get_db_connection():
    """Returns a connection to the PostgreSQL database."""
    return psycopg2.connect(
        host=POSTGRES_HOST,
        port=POSTGRES_PORT,
        user=POSTGRES_USER,
        password=POSTGRES_PASSWORD,
        database=POSTGRES_DB
    )

# Define the Vector

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