Resize your partitions so you can use the empty space.

Then install to the new position.

Update grub.

The real question is: Why?

Here's some answers to your question:

1. Backup your data, nuke the drive, start again.
2. chroot is the tool for the job. Backup your data before you accidentally nuke your drive.
3. Shrink the partition and write a new one in the empty space, but backup your data before you accidentally nuke your drive.
4. Connect an external drive, install on it, but backup your data before you accidentally nuke your drive.

Finally, the reason that the wiki didn't help is that the question is asked by either a person with not enough experience, or one who doesn't need the wiki. This is a non trivial process and you should backup your data before you proceed, lest you accidentally nuke your drive.

Edit:

Also, anything you break can be fixed .. but only if you have a backup.

First step is to shrink and possibly move around your existing partitions to make space for a new one. GParted Live may help with that. Once you have enough space, just install Arch there. If you already have separate /home you should be able to reuse it.

I think it's a messy idea, you will be getting conflicts on files already present in the system. You've been warned ;)

With that out of the way, I guess just download the image and start from https://wiki.archlinux.org/title/Installation_guide#Installation

It's easier with endeavourOS or Cachy OS if you don't want vanilla. As someone who's an Arch user, endeavourOS is better.

Resize your drive so you habe enough space for arch. Then boot into the live USB, partion the empty space into the typical boot/swap/root and then install arch as normal.

resize2fs or a bootable gparted stick will help. And yes, it's all on the Arch Wiki.

You have to create a second partition or use a Virtual Machine. Look for dual booting instructions, if you want to install alongside your current distro.