chore: refine routing in vercel.json and set consistent base paths in vite.config.ts and router.tsx

apps/docs/src/router.tsx

@@ -4,6 +4,7 @@ import { routeTree } from './routeTree.gen';
 
 export function getRouter() {
   return createTanStackRouter({
+    basepath: '/docs',
     defaultNotFoundComponent: NotFound,
     defaultPreload: 'intent',
     routeTree,

apps/docs/vite.config.ts

@@ -10,6 +10,7 @@ export default defineConfig({
   server: {
     port: 3000
   },
+  base: '/docs',
   plugins: [
     mdx(await import('./source.config')),
     tailwindcss(),
@@ -21,6 +22,7 @@ export default defineConfig({
     // see https://tanstack.com/start/latest/docs/framework/react/guide/hosting for hosting config
     // we configured nitro by default
     nitro({
+      baseURL: '/docs'
       // routeRules: {
       //   '/': {
       //     proxy: 'https://classroomio-com.vercel.app/'