We purchased a vehicle, paying the full amount by check. How do I record this transaction in QB, to reflect the expense and the added asset?

Enter on the Check Expense screen that you bought a Fixed Asset, and not something like Office Supplies. But make sure you know if you need to split that entry, because annual registration is an Expense, not part of the Fixed Asset basis (cost).

Stop worrying about JE. You are making Real Transactions, not JE.

You enter the Banking Spending; this is the event that has Payee. It has What you spent. It has the details of what that paid for. This is the Only Entry you need to make. Behind the scenes, it Debits what you entered for the positive values and it Credits Banking (the source of the funds you spent). And you Never use JE for this, because you Spent from Banking.

Banking = checks, deposits, Transfers.

A JE is not the right entry  for anything with Names. Not for Banking, AP, AR, QB Payroll, Sales, Sales Taxes, never for QB Inventory. You are paying for a program that has functions for Names and products/services, for customer and vendor activity. That is Never a JE.