How Does Cramer's Rule Relate to Geometry?

[member 428835]

hey pf!

so my question is how cramer's rule makes sense from a geometric perspective. I'm reading the following article:

http://www.maa.org/sites/default/files/268994245608.pdf

and i am good with the logic of the entire article except one point: when they say $$x=\frac{ON}{OQ}$$ can someone please take a quick second and explain to me why this is the case? i thought from the coordinate transformation we would simply have $$x=ON$$

let me know what you think! i'd really appreciate it!

also, i do hate directing you all to another link but it is too much to put on this post, although it is pretty simple stuff.

 

[fzero]

He is using slightly confusing terminology when he refers to "unit (basis) vectors," which he elaborates on in the Note on p. 36. The vectors $(a,c)$ and $(b,d)$ are being called unit vectors because their lengths define the units in the new coordinate system relative to the old one. Their lengths are not assumed to be equal to 1 in the old coordinate system. So when we compute $x$, we want to do it in the new units, which leads to $x = ON/OQ$. In the old units, it is indeed given by $ON$.

 

[member 428835]

sorry to bring this up again, but rethinking this paper, how is it we allowed the points (a,c) and (b,d) to be paired. in other words, why not (a,b) and (c,d)?

sorry it has been so long, but i am very curious here.

thanks!

 

[chogg]

This doesn't relate directly to the paper you're reading. But if you want a good explanation of the geometrical meaning of Cramer's Rule, check out "Geometric Algebra for Computer Science," by Dorst et al.

Section 2.7.1 explains it rather nicely. If you're not already familiar with bivectors and the outer product, the rest of Chapter 2 gives a good intro.

The basic idea is that the coefficients are just a ratio of areas in the plane.