How are the determinations of fate either adhered to or meddled with in Act 3 of Macbeth?

Fate is defined as the occurrence of events which are beyond one's control and is generally seen as being determined by a supernatural power. It is also believed that one is destined, from birth, to become one or the other and that one's destiny is, therefore, predetermined. Others, though, believe that our destiny is in our own hands and we determine our own fate.

In Act 3, Macbeth clearly wants to further control his own destiny. He has already been successful in achieving his main ambition, to become king. He now wishes to ensure that he retains his position. For him to succeed, he has to remove all threats. He has concluded in scene one that Banquo is a threat, and he muses:

To be thus is nothing;
But to be safely thus.--Our fears in Banquo
Stick deep; and in his royalty of nature
Reigns that which would be fear'd: 'tis much he dares;
And, to that dauntless temper of his mind,
He hath a wisdom that doth guide his valour
To act in safety. There is none but he
Whose being I do fear:...

Macbeth believes that as long as Banquo is alive, his position is not safe. He realises that Banquo would not act in an irresponsible manner and he fears that he, because of his noble qualities, will easily achieve the assistance of others and would, therefore, be more than capable of overthrowing him. He must, therefore, get rid of Banquo. Living in fear of a threat is not living at all.

The three witches had also predicted that Macbeth would be king but that Banquo's heirs would become kings, implying that Macbeth will leave behind a barren throne. Earlier in the scene, Banquo had also alluded to this fact:

Thou hast it now: king, Cawdor, Glamis, all,
As the weird women promised, and, I fear,
Thou play'dst most foully for't: yet it was said
It should not stand in thy posterity,
But that myself should be the root and father
Of many kings.

It is obvious from the above that Banquo suspects Macbeth of Duncan's murder. He also mentions that Macbeth would not retain his power forever since he, Banquo, would be "the root and father of many kings."

Macbeth believes it imperative that he should change Banquo's destiny by killing him and also ensure that Banquo's bloodline ends with him, to prove the witches wrong. He would then truly be the master of not only his own fate but also of others. For this purpose, both Banquo and his son, Fleance, must be assassinated. Macbeth hires three murderers to perform this gruesome task. He has already determined when and where the deed is to be performed since he had asked Banquo details about his plans for that day. 

In scene 3, the attack on Banquo and Fleance, on their return, is described. The assassins manage to kill Banquo, but Fleance escapes. Macbeth's plan has not been completely successful. In scene 4, we read about his frustration when he receives tidings of this failure.

First Murderer
Most royal sir,
Fleance is 'scaped.

MACBETH
Then comes my fit again: I had else been perfect,
Whole as the marble, founded as the rock,
As broad and general as the casing air:
But now I am cabin'd, cribb'd, confined, bound in
To saucy doubts and fears. But Banquo's safe?

Macbeth expresses his anxiety that Fleance managed to get away. He believes that he would have been secure in the knowledge if the entire threat to the legacy he wishes to leave behind in the form of heirs, had been removed. He now has to contend and live with the witches' prophecy that he will leave behind an empty throne. The thought brings him great anguish and perturbation because the three sisters' forecast might just come true as it had with him. Not only that, but Fleance also poses a threat for he will grow up and want to avenge his father's murder.  

Scene 5 conveys Hecate's, (the queen of witches) anger at the three sisters for having manipulated Macbeth without involving her. She suggests a wicked scheme to lead Macbeth further to his doom:

...Great business must be wrought ere noon:
Upon the corner of the moon
There hangs a vaporous drop profound;
I'll catch it ere it come to ground:
And that distill'd by magic sleights
Shall raise such artificial sprites
As by the strength of their illusion
Shall draw him on to his confusion:
He shall spurn fate, scorn death, and bear
He hopes 'bove wisdom, grace and fear:
And you all know, security
Is mortals' chiefest enemy.

All things considered, it seems that Macbeth is, to a large extent, master of his own fate. There are certain things he can change but others he cannot, as in the situation mentioned above. He succeeded in usurping Duncan's throne and murdering Banquo, but he failed in killing Fleance. It seems that Shakespeare is suggesting that, in the end, we are responsible for our own evil and our own destruction, but generally fail in so easily determining the destiny of others, especially if they are on the side of good.