The queen vs rook endgame can be broken down into three steps.
1. Breaking the third-rank defence, forcing your opponent into the second-rank defence.
2. Breaking the second-rank defence, reaching a Philidor position.
3. Winning the endgame from Philidor's position.
This page covers the first step, breaking the third-rank defence.
We'll start from this position, as analysed by grandmaster J.Nunn.
Black is defending with his rook along the third rank.
You might need some help to break this defence, because not all moves are obvious.
Especially the first move may be difficult to find on your own.
What does this move accomplish? It isn't a check or a very forcing line.
Well, this queen move leaves the black rook with no safe squares on the third rank. Therefore it forces the king to move.
2....Kc7 (Kd8 fails to the skewer Qa5); 3. Qa7+,
This move forces the black rook to the second rank. The third-rank defence now changes to a second-rank defence.
Now White will chase the black king onto the last rank and into the corner.
4....Kb8; 5. Kd6,
White is cautious. A move like Kc6 would have lost the queen.
In this kind of positions, you'll want to watch out for stalemate as well.
5....Rg7; 6 Qb4+,
White wants to win the rook, or chase the king into the corner.
Black has succeeded to keep his king out of the corner.
Now White harasses the rook, so his king can eventually go to the best possible square (c6).
8....Ka7; 9. Qd4,
If you already know how to break down the second-rank defence, you may have seen this manoeuvre before.
There's not much to say here. It's a pretty picture, especially if you're White (and you know you're winning).
Here White has achieved a winning Philidor position.
You're not familiar with this position?
You can discover the winning method for the Philidor position in the first section on queen vs rook endgame.