tow forcess F1=(3.0)in the x direction minus(4.0) in the y direction and F2= (-6.0)in the x direction minus (4.5) in the y direction are applied to a particle. what third force would the the net, or resultant, force on the particle zero?

Answer 1

F1(3,-4)
F2(-6,4.5)

Resultant R is the sum of the component, namely R(-3, 0.5). Resultant is the single force equivalent to the vectorial sum of the other two.
If you are looking to have a third force to keep the particle in equilibrium, you would apply an equal and opposite force to the resultant, namely F=F(3,-0.5).

It is not clear from the question if you are asking for the third force that puts the particle in equilibrium, or the resultant of the two.

Answer 2

A body is at equilibrum under the action of three forces one forces 10N acting dew east and one is 5N In direction 60 north east.what is the magnitude and direction the third