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I would like to clip a vector layer containing buildings by a rectangular boundary (also a vector layer). However, rather than cutting straight through the buildings at the boundary, I want it to remove buildings that are fully outside the boundary. Meaning the clipping result I would like is in a jagged boundary, rather than a clean trim.

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The tool you want is called 'extract by location'. You can access it by typing in the search bar in the bottom left hand corner.

Then fill out the form with your buildings layer as extract features from layer and the boundary as the By comparing to the features from layer.

you probably want intersect as your predicate -see the docs for more on what they mean.

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Use the "Extract by location" with either intersect or overlap predicate.

For more details check the Understanding Join Attributes by Location in QGIS?

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