How did union forces eventually defeat vicksburg?

The chief obstacle was how to cross the river unobserved by the commander of the Vicksburg garrison, General John C. Pemberton. Grant achieved this through brilliant deception tactics, ordering a cavalry raid down through the whole length of the state of Mississippi, which distracted Pemberton's attention long enough for him to get the whole army across before Pemberton realised what was happening.after this, the biggest factor in Grant's favour was the long-running feud between Pemberton's area commander, Joe Johnston, and the Confederate President, Jefferson Davis, whereby Pemberton had to try to follow conflicting orders. Davis had told him to hold Vicksburg at any cost, as it was the last Confederate stronghold on the Mississippi, while Johnston could see that it was bound to fall, and urged him to evacuate the place and save his army.In the event, he took part of his army out, in order to bring Grant to battle, but was driven back into his lines, to face a long siege by Grant, which resulted in the inevitable surrender. i hope this helps