To determine the position of a 10kΩ potentiometer with a 5kΩ load when the voltage across the load is 3V, it's essential to understand that the potentiometer divides voltage based on the ratio of its two resistive parts. The resistance can be expressed as x*10kΩ for one part and (1-x)*10kΩ for the other, where x represents the fraction of the potentiometer used. The output voltage can be calculated using the voltage divider formula, which incorporates the parallel resistance of the load and the potentiometer's resistances. The discussion emphasizes the need to derive equations that relate the known voltage and resistances to solve for the position of the slider on the potentiometer. Understanding these concepts is crucial for accurately determining the potentiometer's setting in the circuit.