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```python
# AI Dev Tool: FastAPI PostgreSQL Vector Search Micro-SaaS Boilerplate
# =====================================================================
# This is a high-demand, production-ready developer code asset for building
# a FastAPI-based micro-SaaS application with PostgreSQL vector search capabilities.

from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from pydantic import BaseModel
from typing import List, Optional
import psycopg2
from psycopg2 import Error
from psycopg2.extras import DictCursor
import logging

# Initialize the FastAPI application
app = FastAPI(
    title="FastAPI PostgreSQL Vector Search Micro-SaaS Boilerplate",
    description="A high-demand, production-ready developer code asset for building a FastAPI-based micro-SaaS application with PostgreSQL vector search capabilities.",
    version="1.0.0",
)

# Define the PostgreSQL database connection settings
POSTGRES_HOST = "localhost"
POSTGRES_PORT = 5432
POSTGRES_USER = "postgres"
POSTGRES_PASSWORD = "postgres"
POSTGRES_DB = "postgres"

# Define the PostgreSQL vector search extension settings
VECTOR_SEARCH_EXTENSION = "pg_trgm"

# Define the logging settings
logging.basicConfig(level=logging.INFO)

# Define the data model for the vector search query
class

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