A few important bits this left out (as far as I read, which wasn't too far). Note: I am not a J expert, just a dabbler.

J's tacit syntax can generally transparently take either a single argument on the right, or two arguments, one on the left and one on the right.

In addition to the fork described in the article, J defines a "Hook" for two verbs (instead of the fork's three). A hook applies the right verb to the right argument, then applies the left verb with the original argument on the left and the result from the right verb on the right. Meaning:

   this gets the largest item from a list >./ 
   this divides the left by the right %
   so this scales every item in a list, so the largest becomes 1 and everything else becomes its ratio to the largest: %>./

J allows arbitrarily long strings of verbs: these get forked and hooked until you go insane trying to track it all in your brain.

Defining a function longer than a fork and using the same code inline can (often?) not give the same results. I think that's why the caps are needed in the magnitude function in the article.

I think the article is missing a trick on the magnitude-of-a-vector bit: J has a marvelous conjunction called "under" which, when applied to two verbs, first applies the first (right) verb, then applies the second verb to the result, and then unapplies the first verb.

So when you have the need to "sqrt the sum of the squares" you should immediately be noticing that sqrt and square are opposites, and be thinking "under".

   Under is &.:
   Sum is +/
   Square is *:

So magnitude can be expressed more succinctly (and I think more idiomatically, but again I'm not an expert) as:

magnitude =: +/&.:*:

magnitude 3 4

   5

magnitude 3 4 5

   7.07107