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

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```python
# AI Dev Tool: FastAPI + PostgreSQL Vector Search Extension Micro-SaaS Starter
# ==============================================================================
# This is a production-ready starter boilerplate for building micro-SaaS applications
# using FastAPI, Pydantic v2, and PostgreSQL Vector Search Extension.

from fastapi import FastAPI, Depends
from fastapi.responses import JSONResponse
from pydantic import BaseModel
from typing import List
from psycopg2 import connect
from psycopg2.extras import DictCursor
from psycopg2.errors import UniqueViolation

# Define the database connection settings
DB_HOST = "localhost"
DB_NAME = "mydatabase"
DB_USER = "myuser"
DB_PASSWORD = "mypassword"

# Define the FastAPI application
app = FastAPI()

# Define the Pydantic model for the vector search query
class VectorSearchQuery(BaseModel):
    query: str
    limit: int = 10

# Define the Pydantic model for the search result
class SearchResult(BaseModel):
    id: int
    title: str
    description: str
    vector: List[float]

# Define the database connection function
def get_db():
    try:
        conn = connect(
            host=DB_HOST,
            database=DB_NAME,
            user

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