Answer:
![\vec{F} = -0.34\^x - 0.22\^y\\|\vec{F}| = -0.41~N](https://tex.z-dn.net/?f=%5Cvec%7BF%7D%20%3D%20-0.34%5C%5Ex%20-%200.22%5C%5Ey%5C%5C%7C%5Cvec%7BF%7D%7C%20%3D%20-0.41~N)
Explanation:
The electric force between two point charges can be calculated by Coulomb's Law:
![\vec{F} = \frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}\^r](https://tex.z-dn.net/?f=%5Cvec%7BF%7D%20%3D%20%5Cfrac%7B1%7D%7B4%5Cpi%5Cepsilon_0%7D%5Cfrac%7Bq_1q_2%7D%7Br%5E2%7D%5C%5Er)
We have to calculate the distance between two points; (0,0) and (0.3 m, 0.2 m).
![r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(0.3)^2 + (0.2)^2} = 0.36~m](https://tex.z-dn.net/?f=r%20%3D%20%5Csqrt%7B%28x_2%20-%20x_1%29%5E2%20%2B%20%28y_2%20-%20y_1%29%5E2%7D%20%3D%20%5Csqrt%7B%280.3%29%5E2%20%2B%20%280.2%29%5E2%7D%20%3D%200.36~m)
Now we can apply Coulomb's Law
![F = \frac{1}{4\pi\epsilon_0}\frac{(3\times 10^{-6})(-2\times 10^{-6})}{(0.36)^2} = -0.41~N](https://tex.z-dn.net/?f=F%20%3D%20%5Cfrac%7B1%7D%7B4%5Cpi%5Cepsilon_0%7D%5Cfrac%7B%283%5Ctimes%2010%5E%7B-6%7D%29%28-2%5Ctimes%2010%5E%7B-6%7D%29%7D%7B%280.36%29%5E2%7D%20%3D%20-0.41~N)
The minus sign in front of the force means that the force is attractive.
The direction of the force can be calculated as follows:
![F_x = F\cos(\theta)\\F_y = F\sin(\theta)](https://tex.z-dn.net/?f=F_x%20%3D%20F%5Ccos%28%5Ctheta%29%5C%5CF_y%20%3D%20F%5Csin%28%5Ctheta%29)
where θ is the angle between F and the x-axis. This angle can be calculated by the triangle with edges 0.3 m, 0.2 m, and 0.36 m.
So, sin(θ) = 0.2/0.36 = 0.55 and cos(θ) = 0.3/0.36 = 0.83.
Finally,
![F_x = -0.41 \times 0.83 = -0.34~N\\F_y = -0.41 \times 0.55 = -0.22~N](https://tex.z-dn.net/?f=F_x%20%3D%20-0.41%20%5Ctimes%200.83%20%3D%20-0.34~N%5C%5CF_y%20%3D%20-0.41%20%5Ctimes%200.55%20%3D%20-0.22~N)