Answers for "laravel what permissions vendor should have"

laravel folder permission

cd /path/to/root;
sudo chmod -R 777 ./ 
sudo chown -R www-data:www-data ./
sudo usermod -a -G www-data ubuntu
sudo find ./ -type f -exec chmod 644 {} \;
sudo find ./ -type d -exec chmod 755 {} \;
sudo chgrp -R www-data storage bootstrap/cache
sudo chmod -R ug+rwx storage bootstrap/cache

You don't have write permissions for the rvm

sudo chown -R user ~/.rvm

what Permissions do I need for include folder on php

Directories must have execute permissions to be usable;
Usually this is 0755.
PHP scripts run via mod_php are not executed but rather read;
0644 will suffice for this.