Quaternion rotation matrix unexpectedly has the opposite sense?

This isn't a quaternion problem so much as it is a usage and/or documentation issue with the DirectX call you're using. The transformation the call gives is the one that happens when you move the camera. If you're keeping the camera fixed and moving the world, you're swapping what's moving and what's fixed.

These coordinate transformations are inverses of each other, which is why taking the inverse works for you.

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