If the net force acting on a laboratory cart as is it being pushed down the hallway is zero, then what is the relationship between Fa and Ff?

Fa = - Ff provided that the angle between Fa and Ff is zero

Explanation:

Since the net force is zero, the sum of adding all external forces onto the cart must be zero.

Assuming there is no other forces acting on the cart and Fa acting in the same plane as Fc (i. e. zero angle between acting forces),

Fa + Ff = 0

Hence

Fa = - Ff