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The concept of direction in relation to gravity is complex.

Gravity is a property of matter and can be visulised as a force pulling towards the centre of mass of that matter.

Thus YOU as a body of matter have a mass and a gravitational force of your own.

However, on the Earth, your mass is much smaller than Earth's so you feel Earth's gravity pulling you towards its centre.

On the Moon you woud feel yourself being pulled towards the centre of mass of the Moon.

Thus as you can see what happens applies depends on where you are - but - it is always towards the centre of mass.

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