Find the electric field intensity at point D

1. The problem statement, all variables and given/knownThe diagram shows three small charges at three corners of a rectangle; charge A is -12µC, charge B is -15µC and charge C is +8.1µC. Calculate the magnitude and direction of the electric field intensity at the fourth corner, D.

The Attempt at a Solution

I actually did this entire question with components and got the right answer. However, I find that it is time-consuming, and I was wondering if any of you PF experts know a faster method of doing this.

Components Method:
Split every diagonal vector to x and y vectors and add up all x and y vectors from the three electric field intensity vectors in the image to get the Net[X] and Net[Y] vector. Then use Phythagorean Theorem to find the electric field intensity vector at D, and then use arctan(- / - ) to calculate the angle.

Is there a faster way of accomplishing this?
Thanks,
aeromat.

Related Introductory Physics Homework Help News on Phys.org
Well you really don't have to break it up into components, you can just use vector addition. Its really the same thing but it might save a bit of time. For example if you found the three electric field vectors due to the charges to be

$$\vec E_1 = e_{1x} \hat i + e_{1y} \hat j$$

$$\vec E_2 = e_{2x} \hat i + e_{2y} \hat j$$

$$\vec E_3 = e_{3x} \hat i + e_{3y} \hat j$$

then you can add these to give the resultant E vector

$$E_r = (e_{1x} + e_{2x} + e_{3x} ) \hat i + (e_{1y} + e_{2y} + e_{3y} ) \hat j$$

And then use the same procedure to get the intensity and angle.
It may save a bit of time, and it is good to get accustomed to using vectors.