Luke Skywalker destroyed the Death Star soon after Obi-Wan Kenobi brought him to it but how did Darth Vader know that Luke was his son?

Answer 1

Darth Sidious said that Luke was the Off-Spring of Anakin Skywalker; Vader's alter-ego.

Watch The Empire Strikes Back. Palpatine says , "This boy is the offspring of Anakin Skywalker". Vader asks, "How can that be?"

Vader knew this kid was strong in the Force and intrigued by him, perhaps to grab as an apprentice. But when he learns Luke is his son....well....that's the rest of the story.

Actually, here's the real quotes.

Palp "There is a great disturbance in the Force."

Vader"I have felt it."

Palp "We have a new enemy: the young rebel who destroyed the Death Star. I have no doubt this boy is the offspring of Anakin Skywalker."

Vader "How is that possible?"

Palp "Search your feelings, Lord Vader. You will know it to be true. He could destroy us."

Vader "He's just a boy. Obi-Wan can no longer help him."

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Between Episodes IV and V Luke does more things that garner the attention of the Empire and eventually they find out his name is Luke Skywalker. There is only one Skywalker family in the whole Galaxy so when Vader heard his old name he knew that it was his son and began to plot to turn him to the dark side way before the Emperor talked to him in Episode V. He wanted this so they could both take down the Emperor and control the Galaxy together. Vader always had the intention of overthrowing the Emperor from the very start right after he turned to the dark side. But after his powers were diminished because of his battle with Obi-Wan he became dependent on the Emperor and he wasn't powerful enough to do it himself so he would need an apprentice and who better than his own son.