Feeling lost ? Here's a map of the known universe as in the Star Trek series. A pretty cool map, but it doesn't explain the "Quadrant" thing very well. I guess they defined the transition from Alpha to Beta quadrants as the line straight out from the galactic core that intersects with Sol. (In basically the same manner as defining the Prime Merdian to be the line running from North Pole to South Pole passing through Greenwich, England.) Thus, the Alpha and Beta quadrants are the area where most of the series takes place, because Sol is right on the border between them.
The borders for the Gamma and Delta quadrants are, as a result, the 90 degree perpendicular lines from that line, also starting at the galactic core. Since Sol and its neighbors are out in the arms, this makes the Gamma and Delta quadrants literally on the opposite side of the galaxy.
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