How to find the constants for a set of vectors when a grid is given?

Jun 2017
Lima, Peru
The problem is as follows:

Given the vectors $\vec{A}\, \vec{B}\, \vec{C}$. Find $\frac{mn}{p^2}$ if is known that $m\vec{A}+n\vec{B}+p\vec{C}=\vec{0}$.


I'm stuck on how to use the grid to get the coefficients on the given statement. Can this problem be solved graphically?.

So far the only thing which I could spot was that:


Then by the information given:


From following the logic of the above equation:



But in this given situation there is no way to find all the unknowns as there's not enough equations to solve this system.

Assuming that I'm "eliminating p"



Multiplying by $2$ on the first equation and by $3$ on the second equation:





Then for $p$:



Then by returning to what it is being asked: Which it seems that it turns out that it is not necessary to know the third term in the equation.

$\frac{mn}{p^2}=\frac{11n}{19} \times n \times \left(\frac{19^2}{31^2n^2}\right)$

$\frac{mn}{p^2}=\frac{11n^2}{19} \times \left(\frac{19^2}{31^2n^2}\right)$

Then this results into:


which corresponds to $0.21748$ or the second option to which it does check with the given alternative. But does it exist an eassier method to solve this?.