1. **State the problem:** We have three charges: $q_1 = 6.0 \times 10^{-6}$ C, $q_2 = -6.0 \times 10^{-6}$ C, and $q_3 = 3.0 \times 10^{-6}$ C, positioned as described with distance $a = 2.0 \times 10^{-2}$ m. We want to analyze the forces involving these charges. 2. **Identify distances between charges:** Charge $q_3$ is located such that the distances to $q_1$ and $q_2$ are both $\sqrt{2} a$ due to the right triangle formation. This is because it is positioned diagonally from both $q_1$ and $q_2$ at a distance $\sqrt{a^2 + a^2} = \sqrt{2} a$. 3. **Calculate force magnitudes:** The electrostatic force between two point charges obeys Coulomb's law: $$F = k_e \frac{|q_i q_j|}{r^2}$$ where $k_e = 8.99 \times 10^9$ N·m²/C². - Compute $F_{13}$ (force on $q_3$ due to $q_1$): $$F_{13} = 8.99 \times 10^9 \times \frac{6.0 \times 10^{-6} \times 3.0 \times 10^{-6}}{(\sqrt{2} \times 2.0 \times 10^{-2})^2} = 8.99 \times 10^9 \times \frac{18 \times 10^{-12}}{2 \times (2.0 \times 10^{-2})^2}$$ Calculate denominator: $$(\sqrt{2} \times 2.0 \times 10^{-2})^2 = 2 \times (2.0 \times 10^{-2})^2 = 2 \times 4.0 \times 10^{-4} = 8.0 \times 10^{-4}$$ Thus: $$F_{13} = 8.99 \times 10^9 \times \frac{18 \times 10^{-12}}{8.0 \times 10^{-4}} = 8.99 \times 10^9 \times 2.25 \times 10^{-8} = 202.3 \text{ N}$$ - The force $F_{23}$ from $q_2$ to $q_3$ has the same magnitude since $|q_2|=|q_1|$ and the distance is same: $$F_{23} = 202.3 \text{ N}$$ 4. **Determine directions:** - $F_{13}$: Since $q_1$ is positive and $q_3$ is positive, the force is repulsive, so $F_{13}$ on $q_3$ is directed away from $q_1$, diagonally up-right. - $F_{23}$: $q_2$ is negative and $q_3$ is positive, so the force is attractive on $q_3$ towards $q_2$, diagonally down-left. 5. **Calculate net force on $q_3$:** The two forces are perpendicular (along x and y axes directions). Both have magnitude $F=202.3$ N. The net force magnitude: $$F_{net} = \sqrt{F_{13}^2 + F_{23}^2} = \sqrt{202.3^2 + 202.3^2} = 202.3 \sqrt{2} = 286.1 \text{ N}$$ Direction is at 45° to either axis (since forces equal), pointing diagonally up-right. **Final answer:** The net electrostatic force acting on charge $q_3$ has magnitude approximately $286$ N directed diagonally up-right at a 45° angle to the x-axis.