Hm, redpill relationship advice. Haven't been in any real relationship (the 'relationship' I did have was long distance and she was crap, so I don't think it counts) so I can't attest to its validity. Definitely feels right, though. Obtaining relationships are the focus of a womans life, not a (healthy) mans. That said?
More or less. Chemistry happens because of it. Women love a challenge. Aang is rather easy to tell and easy to read. Woman are much more sensitive to these kinds of things than guys are. And Aang is very obvious with his interest. There is no mystery about his feelings and makes him kinda boring. So it suddenly starts pulling back that interest and focuses on something else, Katara is going to take notice. The thing is, it has to be genuine. If Aang only pulls back just to get Katara to notice him, its clear feint and she'll see it for what it is. Not genuine. How to get Aang to listen to that has always been the problem. Katara in many ways impeded his development in his bending because he was too focused on her. Honestly Roku or the other Avatars should have stepped in and given Aang the Metaphorical slap up the head to get back on mission.
Here though? Zuko is perfect for that role. But Zuko can't be getting with her either, otherwise it will be viewed as self serving and a betrayal of their growing friendship. But I can read the dynamics of this group like an open book already.
Sokka cares about everyone and is acting like the big brother. He's trying to keep the peace and keep them focused on the job. He's doing a fairly good job of it too.
Zuko is still coming out of 'I'm a leaf on the wind and I go where it blows' phase. He's rediscovering his drive and his Fire bending. He's focused on his mission of developing that Fire bending to new heights and stopping the insanity of this war that is tearing his people and this world apart. He is not focused on any girl or relationships to distract him from that goal. Ironically he IS everything Aang should be. And Katara is taking notice of that, which is causing Aangs distress.
Katara is focused on her Waterbending and developing her skills sure. But she's also trying to act as the mother of the group, which is her nature. She is strong willed and cares about everyone. But when it comes to boys she's confused. She's attracted to Zuko here because she's not used to a guys indifference to her. Katara is attractive and knows it. Most pretty girls who become beautiful women do. So she's used to a certain kind of male attention. Aang is easy to read because he wears his heart on his sleeve and he's very obvious about his affections to her. Zuko's indifference confuses her as its not something she's used too. Its a mystery. And women love mystery. Which in the mind of teenage girls makes him the obvious 'bad boy'. In canon he was clearly her enemy and there was a TON of bad blood that hadn't been fully mended even by the end of the show. Here? its serving another purpose.
So getting Aang to man up and focus is key. Him taking attention away from Katara will be noticed quickly. But it has to be him genuinely pulling back. It'll be jarring for her at first as she's think she's done something wrong to hurt him or something. When she learns that isn't the case then her attention will focus on what exactly is going on. Builds a little mystery.
I think Ty Lee or Toph would be better matched with Zuko than Katara or Mei.
The Fire Nation Nobility not to mention his Family would never accept Ty Lee. She's a commoner from the Colonies and the descended from the Air Nomads. Toph is another matter. the Bei Fong family is ridiculously wealthy and influential in the Earth Kingdom. An arranged Marriage to solidify Fire Nation control over a huge swath of the Kingdom is something that I could see happening. It also helps that those two had great interplay and chemistry in the show.
Thus she feels compelled to learn more and get closer to Zuko. She wants to know why he's not interested.
