Yellow lines repersent drain pipe,circles represents air vents leading to the roof and green lines represent my suggested drains.

One of my options is to add a T-shape coupling where the 90 is form at the bath sink and extend the plumbing an extra 2 feet and form a 90 to lead into the kitchen where the washer would be.
OR
Run the washer drain directly to the air vent pipe which is about 3 feet to 3 1/2 feet.
Can I use any of these or they are both incorrect.