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 [itex](a,c)[/itex] and [itex](b,d)[/itex] to be paired. in other words, why not [itex](a,b)[/itex] and [itex](c,d)[/itex]?

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.

 

[blue_raver22]

Hi there,

Thank you for sharing your question and the article you are reading. I can provide some insight on the relationship between Cramer's Rule and geometry.

Cramer's Rule is a method for solving systems of linear equations using determinants, which are mathematical quantities that represent the volume of a parallelepiped in multi-dimensional space. In the context of geometry, determinants can be thought of as a measure of the "size" or "magnitude" of a geometric object.

In the article you shared, the authors use Cramer's Rule to solve a system of linear equations that represents the intersection of two lines in a two-dimensional plane. The solution to this system gives the coordinates of the intersection point, which can be interpreted as the point of intersection of the two lines.

Now, in terms of the specific equation you mentioned, $$x=\frac{ON}{OQ}$$ this is derived from the fact that in a two-dimensional plane, the coordinate of a point can be represented as the ratio of its distance from the x-axis to the distance from the origin (0,0). In other words, the coordinate x is the ratio of the distance ON to the distance OQ. This relationship can be seen in the diagram provided in the article (Figure 1).

I hope this helps to clarify the geometric perspective of Cramer's Rule. If you have any further questions, please do not hesitate to ask. Good luck with your studies!