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The easy solution would be to simply give the object a BoxCollider and a Rigidbody and let the Unity physics engine figure out how it lands. But let's assume you have a good reason why you want to use your own physics and thus want to calculate it yourself.
You will have to cheat here a bit, because in many situations there will be two correct solutions. ...
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Turns out the solution to the fundamental problem is to avoid the complicated math and just use an appropriately-configured HingeJoint. When a player grabs an object, set node A to a kinematic body child node of that controller, and set node B to the object being grabbed. Set both paths to empty strings when they let go.
The actual result is a little wonky, ...
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