'Paper is used for writing on' or 'Paper is used for writing', which one is grammatically correct?

• Paper is used for writing on. or Paper is used for writing.

… which one is grammatically correct ?

• Chopsticks are used for eating. or Chopsticks are used for eating with?

• This desk is used for putting the computer. or This desk is used for putting the computer on?

Of the three pairs of sentences, which is/are correct in each case?

Wikipedia [Now at Prezi, EA] [a parallel example at SimpleWikipedia] includes in a brief introduction about paper

Paper, a thin unwoven material made from milled plant fibers, is
primarily used for writing, artwork, and packaging.

Here used for writing, not writing on.

However in an English textbook for Junior Three students in China, when it comes to the usage of some objects, an example

• paper is used for writing on

is given.

So I want to know when a preposition, after an ing-form and possibly a noun phrase in such sentence structures, can be omitted. Why does the textbook give

• [The lock] is used for locking the door.

but not

• [The lock] is used for locking the door with?