Is the following usage correct:
I drove from Los Angeles all the way down to San Diego.
given that San Diego is at the south of Los Angeles? Can it be used for geographical directions?
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It can certainly be used, though it can be interpreted in at least a couple of ways depending on context.
To the basic statement
I drove from Los Angeles to San Diego.
we add down, indicating a direction, and all the way adds that you drove the entire distance.
But all the way can also be used to emphasize the difficulty or extremity of effort to accomplish a task. Depending on context, then, one could simply be stating a fact about the morning's activity, or complaining about having to drive almost three hours— or about having to leave Los Angeles for the likes of mere San Diego.
May I have a cookie, or are you going to make me go all the way across the street to the café?
Down is not always equivalent to south, regardless of map orientation; see Do I travel “up” or “down” to London from north of the city? and Does “coming down” mean “traveling south”?. While the "away from the center" sense of down is not common in American usage (heading out is usual), that would be an Angeleno's perception of the trip. And San Diego is not only south of Los Angeles, but downcurrent and downwind of it as well.