Why are electric field lines also referred to as "lines of force"?
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You wrote the correct equation, with the only thing to be added being that both F and E are vectors. So they point in the same direction.E=F/q
By convention, the electric field vector points towards the negative and away from the positive. As far as I understand it, the decision for why an electric field vector points a certain way is a decision made by the person drawing the diagram
"Think about the force on the charged object question above..."
Are you referring to the equation that you asked for? In that case, the force is proportional to the electric field. Beyond that, I'm unsure how that relates to the direction of the equipotential lines.