I have a bathroom with shower tub on the right and vanity on the left of the toilet, I am planing to install a washer and dryer machine right by the shower tub.
The question is can I hook up the washer machine drain to the lower section of the toilet drain which is 3" and have no problem?

Yes you can. Cut toilet drain pipe, insert 3x2" Y and lead it to the washer location. Install vent and connect with existing vents in the ceiling or attic. Make sure your stand pipe is 36" tall.

You may find it easier to tap into 2" sink or shower drain. Once you open up the floor, snap a picture and post it here. We'll tell you right away how to proceed from this point on...