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```python
# AI Dev Tool: FastAPI PostgreSQL Vector Search Micro-SaaS
# =====================================================
# This is a production-ready developer code asset for building a FastAPI-based Micro-SaaS
# application with PostgreSQL Vector Search Extension.

from fastapi import FastAPI, HTTPException
from pydantic import BaseModel
from typing import List, Optional
import psycopg2
import psycopg2.extras
from psycopg2 import Error

# Define the FastAPI application
app = FastAPI(title="FastAPI PostgreSQL Vector Search Micro-SaaS",
              description="A production-ready developer code asset for building a FastAPI-based Micro-SaaS application with PostgreSQL Vector Search Extension.",
              version="1.0.0")

# Define the PostgreSQL connection settings
POSTGRES_HOST = "localhost"
POSTGRES_PORT = 5432
POSTGRES_DB = "vector_search_db"
POSTGRES_USER = "vector_search_user"
POSTGRES_PASSWORD = "vector_search_password"

# Define the Vector Search model
class VectorSearchModel(BaseModel):
    """The Vector Search model."""
    id: int
    vector: List[float]

# Define the Vector Search request model
class VectorSearchRequest(BaseModel):
    """The Vector Search request model."""
    query: str
    limit: Optional

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