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When rotating an object, how do I snap one of its local axes to the 3D cursor? In other words, how do I make that axis point directly to the 3D cursor?

A practical example: I want to make the center line of a cylinder point exactly to the 3D cursor.

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I'm afraid it isn't possible to set rotation directly to the 3D cursor but you can achieve a similar effect with a number of steps.

First, add an Empty to the scene - this will, by default, be added at the location of the 3D cursor.

Add a Track To constraint to the object you're trying to orient. Set the Target to the newly added Empty. Configure the constraint To and Up settings to set the required orientation.

In the 3D window press Ctrl+A to bring up the Apply menu. Select Visual Transform - this will set the object's rotation, location, etc. to the current 'visual' orientation (ie, that including any constraints).

You can now remove the constraint and delete the Empty and the object should retain its orientation, pointing at the location defined by the 3D cursor's location.

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