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We draw arrows going up first because each electron must enter an empty orbital first before going into the same orbital because of repulsion (two electrons repel). By putting the arrows up before down, it shows how we can minimize repulsion in a specific element.
The arrows are different directions to indicate the opposite electron spins. Although the electrons aren't actually spinning up and down, we can indicate their spins with up and down arrows. As to why we write the up arrow before the down arrow, I believe it is arbitrary seeing as how the direction of the arrows is arbitrary as well (up and down don't refer to the actual directions).
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