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Is there a way to project a vector3 onto a plane?

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I'm aware of being able to do so onto another vector, but I need to be able to do functionally the same along 2 vectors, or a plane, decided by an object other than the one I want to move along the plane. I'm effectively trying to flatten vectors onto a plane. As a practical example, I have a player in first person view who wants to climb a wall. The direction of his camera is what he uses to decide his direction, but his actual movement is limited to be along a plane of the climbable surface.

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