Police say a grey BMW 1 Series was made to come off the A169 by a black Audi TT Quattro at around 2.25pm on Friday, January 29.
The Audi was travelling from Pickering towards Whitby and the BMW was travelling in the opposite direction.
Anyone who saw the incident or witnessed a vehicle being dangerously or carelessly in the area at the time is being urged to call police.
Officers are also particularly keen to speak to the driver of a Ford Transit that was travelling from Pickering towards Whitby and was overtaken by the Audi.
Anyone with any information, including any dash-cam footage, that might help the investigation is asked to email [email protected] or call North Yorkshire Police on 101, quoting reference number 12210048589.
Information can also be passed on anonymously via Crimestoppers on 0800 555 111.