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I have a document in Illustrator CS6. When I created the document, I used the size of 72" x 40.5". However when I switch from inches to pixels as the size, it gives me a size in pixels of of 5184 x 2916. My desired resolution would be 1920 x 1080. I know that it must be a ppi of 26.67 if it were to be a raster.

It would be ideal to have it in this resolution while I am designing. This is for a projection, so I want to be working with the projectors resolution, but the dimensions of the screen.

How do I get it to match so I can do 1920 x 1080, but at 72" x 40.5"?

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You can't. Illustrator always assumes 1px = 1pt (i.e. 72 PPI).

Since you're working in vectors you don't need to change the resolution at all (technically there is no resolution to change, it just so happens that Adobe decided to make pixels equal to points to make things easier). So you need to pick one. Either work in the correct physical dimensions and export at your desired resolution or work in the correct pixel dimensions and ignore physical dimensions completely.

If the reason you want to do this is to have a pixel preview at the correct resolution and work in physical dimensions at the same time then you can use a rasterize effect as a bit of a hacky workaround...

Add a Rasterize effect (Effect → Rasterize...) to the top layer of your artwork (if you're working with multiple layers you can nest these within a single top level layer).

You don't get pixel level snapping with this so you can also set a custom grid to simulate your pixel grid. So if you're working at 200 PPI you set a grid line every 200th of an inch (or 0.005 inch). The rasterize effect won't line up with your grid by default however. To get around that, put a large rectangle behind your artwork and make sure that it's x and y position values begin at 0 (or any whole number, as long as it is outside of your artwork).

Here I have a rasterize effect on my layer and a custom grid set to simulate a pixel grid:

enter image description here

But to be honest, I would just work at the correct size to begin with...

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