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I want to have one material for a bunch of objects and have them generate a different UV for each. This question has been answered here, but I can't find the "Add" node that accepts vectors.

My Add > Converter > Math > Add node only has grey dots, where I need purple ones.

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You want ShiftA> Add > Converter > Vector math instead:

enter image description here

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    $\begingroup$ It's worth noting that the node label will changing depending on what operation you select; as you can see, it's not actually called an "Add" node. $\endgroup$
    – wchargin
    Commented Aug 21, 2013 at 22:39
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You can also simply use a Color Mix node, since vectors (XYZ) are basically the same as textures (RGB) if you stretch your brain a bit. There's nothing stopping you from plugging purple dots into yellow ones almost anywhere. The same goes for the grey dot (values), the only exception is the green one for shaders.

UVs are two dimensional, so they only have X and Y, or R and G (Though other texture coordinates have a Z/B component). Plug the UVs into a shader's color input and you'll see what I mean. This allows us to manipulate UVs just like textures before plugging them into the Vector of the image (or any texture).

UVs are colours

The top images are the standard UVs and an image mapped with them, and the bottom ones are the same UVs mixed with another texture, then used to map the same brick image. Read more about it here: http://adaptivesamples.com/2013/07/31/cif-4-uvs-are-colours/

This is the node setup for the bottom images:

UV manipulation noodles

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