hang up
means the same as hang: "I found his jacket, which was hanging up in
the hallway".
if you hang up or hang up the phone, you end a phone call. If
you hang up on someone you are
speaking to on the phone, you end the
phone call suddenly and unexpected.
You can use hang up to indicate that someone stops doing a particular
sport or activity that they have
regularly done over a long period. For
example, when a footballer hangs
up his boots, he stops playing
football.
hang-up
If you have a hang-up about something, you have a feeling of fear,
anxiety, or embarrassment about it.
hang on
If you ask someone to hang on, you ask them to wait or stop what they
are doing or saying for a moment.
"Can you hang on for a minute?"
If you hang on, you manage to survive, achieve success, or avoid
failure in spite of great difficulties
or opposition. "Manchester United
hung on to take the cup."
If you hang on to or hang onto something that gives you an
advantage, you succeed in keeping it
for yourself, and prevent it from
being taken away or given to someone
else. "The British driver was unable
to hang on to his lead. ... The
company has been struggling to hang
onto its sales force."
If you hang on to or hang onto something, you hold it very
tightly, for example to stop it
falling or to support yourself. "She
was conscious of a second man hanging
on to the rail. ... a flight
stewardess who helped save the life of
a pilot by hanging onto his legs. ...
He hangs on thightly, his arms around
my neck."
If you hang on to or hang onto something, you keep it for a
longer time than you would normally
expect. "You could, alternatively,
hang onto it in the hope that it will
be worth millions in 10 years time.
... In the present climate, owners are
hanging on to old ships."
If one thing hangs on another, it depends on it in order to become
successful. "Much hangs on the
success of the collaboration between
the Groups of Seven governments and
Brazil."
hang out
If you hang out clothes that you have washed, you hang them on a
clothes line to dry.
If you hang out in a particular place or area, you go and stay there
for no particular reason, or spend a
lot of time there. (mainly American)
"I often used to hang out in supermarkets. ... We can just hang out
and have a good time."
All of these, like most idioms, are for informal use.
Best Answer
I think you were mixing and matching two different idioms
Leave you hanging is one idiom that now appears far more used than "hang out to dry" ... if ngram works that is.
https://books.google.com/ngrams/graph?content=leave+you+hanging%2C+hang+you+out+to+dry&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cleave%20you%20hanging%3B%2Cc0%3B.t1%3B%2Chang%20you%20out%20to%20dry%3B%2Cc0
Cambridge dictionary
Leaving someone hanging evokes more clinging to a liferope or a ledge to me...but perhaps it shared the same laundry root ? : )