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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 high-performance 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
from psycopg2 import Error
import logging

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

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

# Define the FastAPI application
app = FastAPI(title="FastAPI Pydantic v2 Micro-SaaS PostgreSQL Vector Search Extension",
              description="A production-ready developer code asset for building high-performance micro-SaaS applications",
              version="1.0.0")

# Define the Pydantic model for vector search queries
class VectorSearchQuery(BaseModel):
    """Pydantic model for vector search queries"""
    query_vector: List[float]
    limit

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