[EDITED]

I apologize if this isn't the appropriate forum. This doesn't require any integration or derivations (to my knowledge), but it does depend on material I had learned in a multivariable calculus course.

Now, there's a bit of a story behind this:

I work in a soil analysis lab and I'm trying to code a calculator that will tell people how much fertilizer to put down to meet our recommendations without under- or overfertilization. We recommend for N-P-K (% nitrogen, % phosphorus, %potassium) in terms of lbs per unit of area. What I want to do is create a calculator where people can put in up to three fertilizers and receive recommendations on how many pounds of each to apply per unit of area.

So this is how I've been modeling this problem:

Recommendation (rec):

$\displaystyle f(x,y,z)=x_0,y_0,z_0$

Fertilizers (ferts):

$\displaystyle g_1(x,y,z)=n_1 x+p_1 y+k_1 z$

$\displaystyle g_2(x,y,z)=n_2 x+p_2 y+k_2 z$

$\displaystyle g_3(x,y,z)=n_3 x+p_3 y+k_3 z$

where $\displaystyle n_i, p_i,$ and $\displaystyle k_i$ are the percents of N, P, and K respectively.

As you can guess, I'm using the x-axis to represent the amount of N, y for P, and z for K.

I've so far solved the problem in two dimensions and now need to extrapolate it to three dimensions. I've edited this thread to ask if this is the right direction rather than ask for help getting started.

Basically, I'm treating everything as vectors. I'm trying to redefine the rec unit vector in terms of the fert unit vectors. I.e. I'm essentially making new coordinate axes out of the fert unit vectors. I'll find the unit vector as components of the fert vectors using the parallelogram law (which I completely forgot about until just now) for each fert vector. Once I've determined that, I can just multiply the components by the magnitutde of the rec unit vector to find the component vectors. Then I'll just need to play with the numbers so I get the fert parameters to match, leaving the magnitude out as the lbs needed to be applied.

Does that make sense?