DA40 Flight Instructor reported ATC gave a vector heading into conflicting traffic resulted in taking evasive action and a NMAC event.

Narrative

On date; I was forced to take immediate evasive action to avoid a mid air collision while receiving vectors and traffic alerts from ATC. I was aboard Aircraft X conducting a flight from the right seat; with the student in the left. We had just finished our maneuvers in the practice area; and were about 5-6nm north of the ZZZ VOR when the cold call to ZZZ approach was made on frequency; stating our position; altitude (3500' MSL); and intentions to land. We were given a transponder code; and shortly after received radar contact and instruction to proceed direct to ZZZZZ to join the arrival for XXL. Not long after this; we were given a vector off of the arrival for spacing; to which we complied. Due to the very high density of aircraft in the area; I told my student to keep his eyes open and remain vigilant. Not long after the initial vector; I saw an aircraft at my two o'clock at the same altitude and was given a traffic advisory from ATC. I responded with traffic in sight; and was told to maintain visual separation while being given a vector of a left turn to heading 140. After initiating that turn; it was obvious that we were being turned directly into the path of this other aircraft (which I believe to be a Piper Arrow or another low wing Piper with retractable landing gear; color unknown). Seeing the aircraft at the same altitude closing at a high rate; I took control of the aircraft; and immediately pulled power to idle and initiated a steep; aggressive dive to clear the aircraft. If I did not see the aircraft closing and take immediate evasive action; we would have had an almost guaranteed mid air collision within short order; as the other aircraft did not seem to be taking action. After getting out of the dive and leveling off at approximately 2800'; my first action was to continue the traffic scan to ensure safety. After safety of flight was assured; I made a call 'Aircraft X Approach' and was given a causal response to remain outside the Charlie. I responded (admittedly with a very stern tone) 'Aircraft X priority'. I got a response and said 'Aircraft X; I just had to take immediate evasive action and reporting a near miss'. The controller's response was something to the likes of 'Sorry that guy made a turn on us; cleared direct ZZZZZ to rejoin the arrival for XXL'. We rejoined the arrival and performed an uneventful landing and taxi back to our ramp. I used this as a training point with the student; enforcing the absolute necessity to trust your own eyes and maintain situational awareness; even when being told otherwise by controllers; as we are all humans and are subject to mistakes."