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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS PostgreSQL Vector Search Extension
# Description: A production-ready developer code asset for building scalable micro-SaaS applications
#              with FastAPI, Pydantic v2, and PostgreSQL Vector Search Extension.

from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from pydantic import BaseModel, validator
from typing import List, Optional
import psycopg2
import psycopg2.extras
import json

# Define the FastAPI application
app = FastAPI()

# Define the PostgreSQL connection settings
POSTGRES_HOST = "localhost"
POSTGRES_DB = "mydatabase"
POSTGRES_USER = "myuser"
POSTGRES_PASSWORD = "mypassword"

# Define the PostgreSQL connection function
def connect_to_postgres():
    """Establish a connection to the PostgreSQL database."""
    try:
        conn = psycopg2.connect(
            host=POSTGRES_HOST,
            database=POSTGRES_DB,
            user=POSTGRES_USER,
            password=POSTGRES_PASSWORD
        )
        return conn
    except psycopg2.Error as e:
        raise HTTPException(status_code=500, detail=f"Failed to connect to PostgreSQL: {e}")

# Define the Vector Search

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