Why does Darth Vader not use the Force to prevent Luke from falling?

In his earlier conversation with Emperor, he says that either Luke will join us or die.

            VADER
    If he could be turned, he would 
    become a powerful ally.

            EMPEROR
    Yes.  Yes.  He would be a great 
    asset.  Can it be done?

            VADER
    He will join us or die, my 
    master.

When Vader asks Luke to join him, he denies that.

            VADER
    There is no escape.  Don't make 
    me destroy you.  You do not yet 
    realize your importance.  You 
    have only begun to discover your 
    power.  Join me and I will complete 
    your training.  With our combined 
    strength, we can end this destructive 
    conflict and bring order to the 
    galaxy.

            LUKE
    I'll never join you!

Also, Luke was wounded and already hurt too much since his friends are captives of the Empire. Vader sees him too weak to be a Sith and since he already refused the offer, so there was no point of saving him. Though, Luke survived and he tries to capture him later.

Another possibility is that he was surprised by Luke's fall, so he didn't get time to use the Force to catch him.

Script Source

It's possible Vader did save Luke.

When Luke lets go, he doesn't fall straight down:

 INT. REACTOR SHAFT

 Suddenly Luke is sucked into an exhaust pipe in the side of the shaft.  
 When Vader sees this, he turns and hurries off the platform.

There doesn't seem to be much air movement in the side vent, certainly not the gale-force wind which would be needed to pull Luke across the width of the main shaft.

There are at least three possibilities:

1. Vader actively saved Luke, by using the Force to push him into the side shaft;

2. Vader knew through Jedi precognition that Luke would survive his fall -- the opposite of "having a bad feeling about this";

3. Vader didn't know if Luke would survive, but regarded the fall as a test of Luke's Jedi skills -- maybe Luke used the Force to push himself into the side shaft.

After Luke is rescued by the Millennium Falcon, Vader wants to capture him:

 INT. VADER'S STAR DESTROYER - BRIDGE

 Vader stands on the bridge looking out the window as Admiral Piett 
 approaches him.

            PIETT
    They'll be in range of our 
    tractor beam in moments, my lord.

            VADER
    Did your men deactivate the 
    hyperdrive on the Millennium 
    Falcon?

            PIETT
    Yes, my lord.

            VADER
    Good.  Prepare the boarding party 
    and set your weapons for stun.

But he is overconfident in his prospects of catching Luke, and didn't expect the hyperdrive on the Millennium Falcon to have been repaired, so Luke gets away.

I submit that it appears to be very difficult , perhaps impossible, for a force user to directly affect the physical person of another force user.

We see force users strike each other with lightsabers, throwing inanimate objects at each other, and hurling energy bolts at one another.

But we don't see for example, Darth Vader force-choking Obi Wan Kenobi. We see young Obi Wan and Anakin Skywalker using force pushes on battle droids, but not against Darth Maul, Count Dooku, etc.

It could be that once a person unlocks their own force use past a certain point, their body can't be directly manipulated by another.

I would like to officially disagree with the accepted answer as I think it cheapens what is to me the most interesting story arc of the original trilogy.

I would posit that this scene found Darth Vader in a moment of weakness as we are just beginning to see the effect that reconnecting with his (when the movie came out) only child is having on the otherwise baddest guy in the galaxy.

A J's answer lists the evidence:

1. Vader swears fealty to the Emperor by swearing to kill Luke if he refuses to join them

2. Not only does he not kill Luke, but he throws out the traitorous suggestion that Luke join him to overthrow the Emperor and take over the Galaxy. (And yes, I am aware that canonically the Sith have established a routine around double-crossing their leaders but out of universe this canon had not been created when the movie was made)

Then to further cement the point home, Royal Canadian bandit adds:

3. Even though the Empire has the Millenium Falcon at a profound strategic advantage, Darth Vader chooses to attempt a capture as opposed to opening fire on the fleeing Millennium Falcon. Clearly attempting to keep him alive as opposed to trying to destroy him (as A J's answer would suggest)

I would like to suggest that Darth Vader could not save Luke because he was not focused at that moment, having just begun to feel that conflict that would eventually turn him back to the Light in Episode VI.

I'm aware that my answer is borderline non-canonical but I feel this is one of those areas where the original drama that made Star Wars so universally successful has been limited by the canonical backstory that has been added over the years to explain certain things. In the end, the original Star Wars story was one of redemption of the big bad villain and to me the possibility that Vader couldn't be completely evil made him all that much more interesting.

Why would he? He want Luke to willingly became his apprentice. When Luke decide to fall it's a sign that he maybe don't want to do that. So DV assume that he may die in the process of falling. And if not, well, it's not like he will suddenly get better at jediing.

There are many possible answers, as can easily be seen from the many replies here offering many reasonable possibilities. So it's mainly a matter of speculation.

I think in the end the "real" answer is simply that that was what the script was asking for. The action, for the script to be successful, had to go the way it went and that's it. A clear explanation for what is happening apparently was not deemed necessary.

You could also ask how it is that the Tie Fighters make a shrieking, howling sound in space, where there is no air? Probably because in the end the effect is what was sought after, and in this case the film makers even went so far as to compromise with known science.

For clarification for what a lot of people was asking (@osboy1); there is a reason why we don't often see force users using force-based attacks on each other (like why they don't continually force push each other). The reason for this is an ability called Force Barrier which is a basic technique whereupon the Force User surrounds themselves with energy that protects them from all but the most devastating force abilities (e.g Force Lightning although Yoda showed an ability to deal with this). This is why you don't see Jedi and Sith taking it in turns to Force Push each other across the room. A force barrier can be broken, though it needs a significant power gap and it takes a lot to break through; most force-users won't even bother trying because it will take too long and/or will be too hard. Of course, fatigue or taking someone by surprise is a way to get someone with their guard down and successfully attack them with the force.

Keeping this in mind, it is possible though not definite that Vader did not have adequate time or ability to break through Luke's force barrier (which he should've been trained to keep active at all times by Yoda or else Yoda's incompetent) to stop him from falling. Luke is naturally talented (and was under immense stress which may have enhanced his abilities) and perhaps Vader needed longer to break through his barrier to actually be able to manipulate his body with the force.

Of course, the most obvious explanation was that he was taken by surprise by Luke's audacity and may have expected him to actually join him after revealing that he was Luke's father. (Remember, Vader was extraordinarily loyal to those close to him, in particular his family and may have expected Luke to be the same.)

Another possible explanation was that it was a test of whether Luke could survive the fall to see if he was a worthy recruit for the dark side or not but I find this less plausible since Vader probably doesn't want him to slip through his grasp.