The purpose of the six day war was to Establish Israeli power in the Israel peninsula and the Gaza strip.
Explanation:
The six day war is also known as the third Israeli- Egyptian - Syrian war named on the main powers that fought the war and was based around the same region of territory which was taken over by Israel in the previous war.
This began with a preemptive strike by Israel and a quick takeover of the disputed lands from the territory of the other two assailants.
The result was the complete control of Israel over the entire peninsula and their undisputed claim over Gaza strip which has continued till now.