STAR WARS

Vader Protects Leia

Over and over, Vader makes sure that the little Princess escapes harm

When an uninformed Imperial officer tells Vader, "She'll die before she'll tell you anything"
Vader replies, "Leave that to me", making it very clear to all that he alone will be dealing with the Princess.
It's only later as the movies unfold that we learn that harming Vader's family is a dangerous enterprise.
Vader is very protective of his children.

First, there's the interrogation itself.
Vader personally oversees this to ensure nothing untoward happens to his daughter
The droid floating into the room is designated a 'Torture Droid' in the screenplay.
Looks like they're juicing her up with some sort of truth serum - 'Sodium Pentothal Delivery Droid' would seem more accurate.
The scene cuts so we're left to only imagine awful things happening - no actual footage of ill deeds.
When next we see her there's no bruising, her clothes are not disarrayed, her hair isn't even mussed.
Torture?
Perhaps 'light interrogation' would be a better description.
With the stakes involved, she's being handled with kid gloves.
Compare this with the Jedi version of 'trade negotiations' in Phantom Menace.

Later, she doesn't flinch at Vader's approach - she seems altogether well composed.
She DOES flinch away from Tarkin, the true villain of the movie.

At the news that the Death Star is operational, Tarkin ponders, "Perhaps she would respond to an alternative form of persuation."
Vader turns on Tarkin and demands, "What do you mean?!"
Vader is alarmed and angry, ready to jump to his daughter's defense

And then Tarkin, furious that she lied to him about the location of the Rebel base, orders Leia's execution.
It's Vader who persuades him to hold off on killing her

When a rescue ship arrives to free the Princess,
It's Vader who tricks Tarkin into letting her go, with the plans that he admits she surely has (seeing as how he's all but delivered them to her)