The general sign for "purpose" is MEANING. Some people "do" sign ON MEANING for "on purpose" -- however, there are often better choices depending on your context. For example:
GOAL: MY GOAL = "My purpose" or "my intention."
REASON: MY REASON DO THAT = "My intention or goal for doing that..."
PLAN: YOU PLAN THAT! = You planned that! = You did that on purpose."
WANT: I WANT ... = "My intention or my goal is ..."

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The sign for "purpose" is the same sign as MEANING. This sign also means "intent."

When done with furrowed eyebrows this sign is used to ask "What does that mean?" or "What is the meaning of _______?" 

MEANING / PURPOSE:

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