I'll create a user authentication webapp using Vue.js, Vite, and deploy it on Vercel. We'll use Vue 3 with the Composition API for this project. Let's break this down into steps and create the necessary files.

Let's start by creating the main files for our project:

Now, let's create our login component:

Next, let's create the registration component:

Now, let's create a simple dashboard component for authenticated users:

Let's create a composable for handling authentication:

Now, let's set up the router:

Finally, let's update the main.js file to include our router and Tailwind CSS:

For Tailwind CSS, create an index.css file in the src directory:

To set up Tailwind CSS with Vite, you'll need to install some dependencies and create a configuration file. Run the following commands in your project directory:

This will create a tailwind.config.js file. Update it to include your Vue components:

Lastly, to deploy this app on Vercel, you'll need to create a vercel.json file in the root of your project:

This configuration tells Vercel to handle client-side routing by always serving the index.html file.

To deploy your app:

1. Push your code to a GitHub repository.
2. Go to Vercel and sign up or log in.
3. Click "New Project" and import your GitHub repository.
4. Vercel should automatically detect that it's a Vue.js project and set up the build settings correctly.
5. Click "Deploy" and wait for the deployment to finish.

Your Vue.js authentication app should now be live on Vercel! Remember that this is a basic example and doesn't include actual backend authentication. In a real-world scenario, you'd want to integrate with a backend API for secure user authentication and data management.

I see you're encountering TypeScript errors while building your Vue project with Vite. Let's address these issues one by one to get your project building successfully.

1. First, let's create a shims-vue.d.ts file in your src directory to declare the Vue file types:

2. Now, let's update your src/main.ts file to use TypeScript:

3. Let's update the src/router.js to src/router.ts:

4. Update your src/composables/useAuth.js to src/composables/useAuth.ts:

5. Now, let's update your src/App.vue file to use TypeScript:

6. Update your tsconfig.json file in the root of your project:

7. Finally, update your vite.config.ts file:

These changes should resolve the TypeScript errors you're encountering. Make sure to install the necessary TypeScript dependencies if you haven't already:

After making these changes, try building your project again:

This should resolve the TypeScript errors and allow your project to build successfully. If you encounter any other issues, please let me know, and I'll be happy to help you resolve them.