Well, well, well, we have ourselves a puzzle to solve! Let's get cracking, shall we?
To find the magnitude of the total electric field E, we need to sum the two parts together. So, we have E = E1 + E2.
Using trigonometry, we can find the x and y components of each electric field vector. The x-component can be calculated as Ex = E * cos(θ), and the y-component can be calculated as Ey = E * sin(θ).
For E1, we have Ex1 = E1 * cos(θ1) and Ey1 = E1 * sin(θ1). Similarly, for E2, we have Ex2 = E2 * cos(θ2) and Ey2 = E2 * sin(θ2).
Now, let's add the x and y components of E1 and E2 together:
Ex = Ex1 + Ex2
Ey = Ey1 + Ey2
By using these equations, we can calculate the magnitudes of Ex and Ey.
Once we have Ex and Ey, the magnitude of the total electric field E can be found using the Pythagorean theorem: E = sqrt(Ex^2 + Ey^2).
Finally, to determine the direction of the total electric field, we can calculate the angle it makes with the +x axis using the inverse tangent function: θ = tan^(-1)(Ey/Ex).
Now, all you have to do is plug in the given values and crunch those numbers. Good luck, math wizard!