How Do You Apply Coulomb's Law to Calculate Forces in a 3D Coordinate System?

[CaptainOfSmug]

Homework Statement

An xyz coordinate system contains three charged particles: particle 1, q1=− 6.5μC , at (4.0 m,-2.0 m, 0); particle 2, q2 = 12μC , at (1.0 m,2.0 m, 0); and an electron at at (-1.0 m, 0, 0).

Determine the x, y, z components of the vector sum of the vectors of the electric forces exerted on the electron by particles 1 and 2.

Homework Equations

F = \frac {kq_1q_2} {r^2_(12)}
Distance formula

The Attempt at a Solution

Okay,

So I first started off by plotting all the points, finding the expected final force and direction pictorially. Now, plugging and chugging Coulombs Law was my first instinct but unfortunately I came into problems (most likely with vectors). I know I needed to find the distance between both the particles and the electron. I did this by just using distance formula where I got:
Distance between 3 and 2: 2\sqrt2
Distance between 3 and 1: \sqrt29

Now here is where my problem is arising. Obviously this is in 3 space so I'm assuming the directional values applied to Coulombs law will need to be in vectors. My book did not go into any sort of detail on this and I'm not sure how the formula would then look.

Could I potentially just take make a unit vector out of all three points and then multiply them by the respective force formulas? And if this is the case, would my distance formula idea be wrong? I'm trying to remember if it is "legal" to multiply my scalar by a unit vector...

Anyhow I greatly appreciate any help, I've been pondering this problem for a couple hours now with not a whole lot to show for it...

Cheers!

 

[Simon Bridge]

The vector form of coulombs law is: $$\vec F_{12} = \frac{kq_1q_2}{r_{12}^3}\vec r_{12}$$ ... where I am using a notation so that: $F_{12}$ is the force on particle 2 due to particle 1, $\vec r_{12}$ is the position vector pointing from particle 1 to particle 2 and $r_{12} = |\vec r_{12}|$

If you have $\vec r_1$ and $\vec r_2$ which are the vectors pointing from the origin to positions 1 and 2 respectively, then what is the vector equation that relates them to $\vec r_{12}$ which is the vector pointing from position 1 to position 2?

In your problem, you are going to have to break the vectors down to their x y z components.